A math-inspired love story in verse

For post #100 on this blog I wrote a poem.  This is post #200 and I decided to start a tradition.

This is a poem written jointly with Allison Bishop.  It is an epic love story.  Math is involved.  The title is still in the works.

The triangle was born in a triangle land
Where lived all triangles, hand in hand
They lived in three-sided houses and went to three-sided schools
They ate off three-sided plates and sat on three-legged stools

But one day the triangle met a square
And was stopped in her tracks by his four-sided stare
And the square himself realized in surprise
He’d never seen such beauty with his four-sided eyes

So their lives together they decided to share
But with this resolved, the question was where?

For the triangle world the square proved unfit,
‘Cause through triangle doors he just could not fit.
After breaking a few when he could not make way,
The other triangles chased him away.

So the triangle followed him to four-sided land
But quickly found that wherever she’d stand
Her pointy end would poke and tear
So the quadrangle world they could not share.

The square had an idea, he would save the day.
He could simply give one of his angles away!
Since he had four angles, and she had three
He could cut one off, and well-matched they’d be

So he took some shears and chopped off a corner
Then looked in the mirror and found to his horror
Where four sides had been, there were now five
Oh what a cruel day to be alive!

But the triangle thought fast, she knew what to do!
If the square could gain one side, then she could gain two!
She picked up the shears, she was not afraid,
With a snip and a snap, two cuts were made.

He looked at her with sadness no more,
So filled with joy that he could soar.
Now they both had five sides, and went off together
To live happily in five-sided land forever.

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A figure with pointy things and a line, a line, and a line

Look at the title.  Did you guess that this was a description of an acute triangle?

This description was from a student in my third grade class.  Let me assure you, this student knows how to recognize an acute triangle and how to draw one.  But verbal descriptions are much trickier for them.

I believe that verbal descriptions are just as important for communicating math as they are in literature, history, science, etc.  That is why we play games and do activities in my math class that practice precisely this skill.

The description in the title came about from one such game that we played.  We had just finished a unit on shapes (mostly focused on polygons) where the students encountered a lot of new terminology about angles, triangles, quadrilaterals, and other polygons.

I gathered all of the terms and wrote each one on its own slip of paper.  The papers were then all placed into a “hat” (this can be a box, a bag, an actual hat, or anything that can store the papers and the students can reach into without seeing the words on the slips).

Play proceeds as follows: two students are called to the front, one to be the “explainer” and one the “guesser” (I made up a schedule so that all the students had an equal number of opportunities to play both roles and so that everyone had a chance to explain to everyone else).  One minute is put on the clock (we tried 30 seconds at first, but that didn’t seem to be sufficient).

When the moderator says “go”, the explainer takes out a piece of paper from the hat at random and attempts to explain the word to the guesser.  The rule is that you can’t use any word or part of word that is written on the paper (so if the term is “right angle” or “triangle” you can’t use the word “angle” in your explanation).

When a word is guessed, the explainer takes out another word and goes on to explain that one.  The explainer continues to take out new slips and explain the words until the time runs out.

If the explainer gets stuck on a word (either because they don’t know what it means, they don’t know how to explain it, or the guesser is just not getting it), they can put it aside and take another one.  The rule, though, is that you can do that with only one word per turn (later I thought that perhaps allowing to put aside 2 or 3 would have worked better).

The first few rounds that we played, no words were guessed.  This was mostly due to descriptions being like the one in the title and the students still getting a feel for the game.

But soon the students started feeling a little more comfortable with what they were doing and putting a bit more thought into their explanations.  Here is what some of the early ones to be guessed sounded like:

“They can be of different types.  A rectangle can be one, a square can be one, and there are many others of these.  Well, you know, with 4 sides.”

Up until the explainer mentioned the 4 sides, the guesser looked very confused, but as soon as the 4 sides were mentioned, the word quadrilateral was said immediately.

Often, concepts were described by what they are not.  For example, “acute angle” was described as “it is not obtuse or right.”  Or “rectangle” might be described as “it is not a square but a…”

But overall, the descriptions got more precise as the game went on.  Some of my favorites were, “It is the shape of a stop sign” (octagon), “It is both a rhombus and a rectangle” (square), “It is a stupid shape with three sides” (obtuse triangle).

What sort of games and activities do you do with your students to practice communicating mathematics?

 

 

 

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Protecting against magic

Yesterday, Zoe and I were playing a game of Bingo.  In this version, there are cards with the pictures and names of the items (all related to the ocean) and you flip them over one by one to figure out which spots to cover on your board.

The cards were all in a pile facing down and Zoe would pick them up one at a time and without showing me have me guess what was on it.  I had to find the correct item on one of our boards (or say that it wasn’t on either of them).

The first time that I pointed to the correct item, Zoe looked quite surprised and said, “yes, how did you guess?”  Well, one time can be a coincidence, right?  But then she did it several more times and for every one of them I guessed correctly what was on the card.  Now Zoe was very intrigued.

What Zoe didn’t realize, is that the names of the items (along with their descriptions) were written on the backs of the cards.  So even though I couldn’t see the picture, I could still identify the item based on the name.

I told Zoe that I could see through the card and tell what was on the other side of it.  Zoe seemed a bit skeptical, but finding no other explanation she bought into this one.

To correct for this phenomenon, Zoe began picking up the cards in such a way as to cover the back of the card right away and keep it covered until I made a guess.  But this didn’t seem to change much – I still guessed all of them correctly.

How? The back of the card would become visible as soon as she’d pick up the previous one, so I had plenty of time to see it before she picked up the card.

But Zoe was not ready to give up.  She put the whole pile of cards behind her and started pulling the cards out from the middle of the deck.  She would then look at the card and put it face down on the floor behind her back.

I thought that this was quite clever of her (especially the part of taking the card out of the middle), but the problem was in the mechanics.  The way she’d leave the card lying behind her, I could still see what was written on the back.

Finally, Zoe resorted to desperate measures – she made me close my eyes while she was selecting the card and would hold it in such a way that I couldn’t see any part of it when I opened them back up.

I was beaten and my magic powers of guessing the item disappeared.  Later she took pity on me and would give me clues about the item, which turned it into a totally different game entirely.

Zoe never figured out how I was magically guessing what was on her cards but I was impressed with her persistence to figure out a way to prevent me from doing it.

I think that one day, perhaps several years from now, Zoe will be playing or looking through the game and have a great revelation.  But I don’t mind if she thinks that I have magic powers for just a little while longer.

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Skip counting or word skipping?

We like games at our house, but there are rules that one has to follow when using them.  And the rules are not just for playing the games (those we actually often modify at will), but there are also rules for putting it away (which are stricter).

One of the rules when putting away the game BLOKUS (a favorite) is that you have to count the pieces and make sure that there are 21 of each color.  Today, after playing with the game, Zoe (4.5 yo) diligently picked up a bag and started putting blue pieces into it.  She counted normally until 7, but then she put in the next piece and said “nine”.

I naturally thought that she just forgot about 8 and pointed out to her that she skipped it.

“I’m counting by 2’s now,” she said.

“But you put in only one piece,” I countered.

Zoe looked at me with surprise.  Without any words, her look said it all:  “Yes. And?”

“When you count by 2’s,” I explained, “you have to put in two pieces.”

“Oh,” said Zoe, and took out the piece that she had just put in.  Without any questions, she picked up a second piece, put them both into the bag and triumphantly said “nine” one more time.

Zoe then proceeded to count the remaining pieces by 2’s, making sure to put in two pieces each time.

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I was surprised that Zoe didn’t ask me why she had to put in 2 pieces at a time.  It must have made sense to her.  And yet, she was perfectly happy counting by 2’s and putting in one piece before I pointed this out.

I have actually seen this phenomena before.  Kids understand the concept of counting by 2’s and are good at figuring out the sequence of words, but then when it comes to counting physical objects they don’t connect it to counting 2 of them at a time.

I think that Zoe understood the connection this time, but I won’t be surprised if she makes the mistake again before fully internalizing it.

What similar misconceptions have you noticed children having?

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Numicon – combining geometry and arithmetic

Translated from this post by Jane Kats.

This year, we are actively using the Numicon tool in our classes.  We have two sets like this one – and it’s plenty for a group of 12. (Translator’s note – here is a slightly smaller set more readily available on amazon).

Sometimes I ask the children to bring me two pieces of different colors and count how many holes there are altogether.  

And recently, we drew another set of puzzles for Numicon. 

You have to find two different Numicon pieces to make each shape.  (Translator’s note – instructions on the sheet say “Put this shape together using two different pieces of Numicon”).

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You need to pick the pieces, see if they fit and count the holes.

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It’s interesting that when kids use the same two Numicon pieces for a different shape, they still have to recount the holes.

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It’s a nice bonus that many of these problems have 2-3 solutions.

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How do you play with Numicon?

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Discussing the meaning of “almost” with an 8 year old

I had previously mentioned that my children are very picky and slow eaters.  This creates much frustration for me, but at the same time has been on multiple occasions inspiration for interesting math conversations.

Yesterday, Katie was eating a cutlet for dinner, and I, as usual, was trying to make her hurry up a bit.

K: I’m almost done with it.

I look at the cutlet and notice that there is still almost half of it left.

Me: What fraction of the cutlet do you think must be remaining for you to be “almost done with it”?

K: I think a third.

Me: Why a third?

K: Because a third feels so much smaller than a half.

Me: Well, you definitely have more than a third left.

K: (taking a big bite) Not any more.

She went back to chewing, very slowly of course, and I thought the conversation was over.  But about 5 minutes later, when she was still not finished with the cutlet, Katie suddenly decided to resume the conversation.

K: Mom, if someone could eat the whole cutlet in just one bite, could they say that they’re almost done with it before they take the bite?

Me: I don’t know.  Can you be almost done with something before you even start?

K: Because if they can’t say it before they take the bite, they can never say it, because after taking the bite they will be fully done with the cutlet!

I couldn’t argue with her there.

So what do you think, does “being almost done with a task” depend on how quickly you do it or just on what fraction of it you have already done?

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From math-hater to mathematician and computer scientist: The story of Allison Bishop

This is the story of an unlikely mathematician. It is the story of my friend and Funville Adventures coauthor Allison Bishop, who always imagined she’d write a book someday, but would not have guessed it would be about math.

Allison grew up with a strong dislike for math.  She was by no means bad at it and had no trouble getting A’s in all her math classes.  She just found it dull, formulaic, and uncreative.  What she loved was writing, where you could use your imagination and create anything you wanted.

It didn’t help that she never cared for most of her math teachers.  There was even one high school math teacher that felt that girls had no business doing math.  When a girl asked a question in his classroom, his reply was, “You will not need to know that when you are a housewife.”

That’s why when Allison began her undergraduate studies at Princeton, math was not anywhere to be seen on her list of potential majors.  She was leaning towards English, although considering some other subjects in the humanities.

Yet somehow, she found herself being convinced to take a math class her Freshman year.  She didn’t have high hopes for it, but thought that she’d give math one last chance.  The class was an Introduction to proofs through Number Theory taught by Jordan Ellenberg.

It didn’t take long for Allison to fall in love with the class.  But how could it be that this was math?  It was so different from anything she had seen called math before.  It was exciting, thought-provoking, and required quite a bit of creativity.  Was this a fluke?  Allison had to try out a few more math classes to be sure.

By the end of her sophomore year, Allison knew that she wanted to become a math major.  She went on to take many more fascinating math classes and to graduate with high honors in math.

She set out to find compelling uses of math in the wider world, and wandered into computer science research as a result. After obtaining her Phd, Allison became a computer science professor at Columbia University. Though she loves teaching, she is not content to work only with students who have already found their way to math and science. She wants to help more young minds discover the creative side of mathematics, and waiting for college is often too late.

Allison joined the Funville Adventures project because she believes that mathematics needs more writers, and writers need more mathematics. The kind of thinking muscles that mathematical studies develop are sorely needed in today’s world, and the creativity that lies behind the shape of a theorem is not so different from that behind the shape of a story. And every little girl, and little boy, should get to grow up loving math. Because in addition to enriching their lives, they will need to know it after all.

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