Multiplication as Scaling with a Multiplying Menace

“If you want your children to be intelligent, read them fairy tales.”

This quote is often attributed to Einstein. Whether he actually said it is perhaps little more than folklore itself.  However, I do love how Jack Zipes transforms the quote into the tiny fairy tale below in Breaking the Magic Spell: Radical Theories of Folk and Fairy Tales. 

Once upon a time the famous physicist Albert Einstein was confronted by an overly concerned woman who sought advice on how to raise her small son to become a successful scientist. In particular she wanted to know what kinds of books she should read to her son.
‘Fairy Tales,’ Einstein responded without hesitation.
‘Fine, but what else should I read to him after that?’ the mother asked.
‘More fairy tales,’ Einstein stated.
‘And after that?
‘Even more fairy tales,’ replied the great scientist, and he waved his pipe like a wizard pronouncing a happy end to a long adventure.

Fairy tales develop the imagination. Imagination has moved fields like science and mathematics in directions once unimaginable. Imagination is magical that way.

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In this week’s picture book, the Brothers Grimm fairy tale character Rumpelstiltskin brings us some multiplication magic.

The Book

The Multiplying Menace: The Revenge of Rumpelstiltskin by Pam Calvert was published by Charlesbridge in 2006 and illustrated by Wayne Geehan.

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In The Multiplying Menace, Rumpelstiltskin returns, demanding repayment for the gold he spun for a princess (who’s now Queen). Rumpelstiltskin requests that the Queen’s son, Peter, works off the debt. Threatening to increase the pest population and decrease the kingdom’s livestock if repayment is not made. Peter must figure out how Rumpelstiltskin’s magical stick works to save the kingdom.

Calvert has created an interesting vehicle for teaching children the differences between multiplying with whole numbers and with fractions.  Geehan’s expressive paintings provide a perfect backdrop for this fairytale, complete with a wholly menacing Rumpelstiltskin.

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The Math

With a magical multiplication stick, Rumpelstiltskin (and later Peter) scales quantities up and down.

 

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What scale factor did Rumpelstiltskin yell out before decreasing the cows from 20 to 4? 

As illustrated in the image above, the book presents the important idea that multiplication can both increase a quantity and decrease a quantity.  Although multiplication by 1 doesn’t appear in book, multiplication by zero does.

This book is about multiplication as scaling. [WARNING: This is a long post and I don’t the time to shorten it. If you are pressed for time, you may want to skip to the MAGIC section and come back to this mathematical discussion later.] While reading, I recalled a set articles written mathematician Keith Devlin in his monthly column sponsored by the Mathematical Association of America.

 It Ain’t No Repeated Addition (In which he refers to his original 2007 article,  What is Conceptual Understanding?)

It Still Ain’t No Repeated Addition

If you teach K-12 math, and you haven’t read these, I’d imagine these are provocative titles. They were/are still for me. While I get Devlin’s point(s), appreciate his explanations and I learned a lot from reading them, I was left with many questions about how to support students.  There are two follow-up blog posts regarding Devlin’s ideas, which I happily discovered recently.

  1. A post by Denise Gaskins  where she unpacks some of Devlin’s ideas.
  2. A post by Christopher Danielson where he experiments with the operation of exponentiation.

In the comments of Danielson’s post, he provides a way of moving forward by asking: What are some mathematically honest stories we can tell about exponentiation [or in this case, multiplication] in relation to [children’s] intuitions [about these operations]?” 

[Note: Danielson means “stories” in the broad sense (e.g., The stories we tell our student about what mathematics is and how it works). Not picture books specifically, however they happen to be a tool for story telling. ]

While the mathematically honest stories/conversations about multiplication that I will and do share with my children/students are still being revised, there are a few underlying themes.

  1. Repeated addition is a useful way to determine answers to multiplication situations when operating with whole numbers. [e.g., When my son asked how to “do” multiplication problems in 1st grade, we talked about 3×2 being 3 groups of 2 (2+2+2).]
  2. Whole numbers aren’t the only types of numbers in mathematics.  Repeated addition will be useful for determining answers to multiplication situations with some types of numbers and not others.
  3. Multiplication is useful for finding the result of scaling some quantity.
  4. Multiplication is an operation with its own web of connected ideas, one of which is repeated addition and another which is scaling.
  5. Multiplication is an abstract operation governed by a list of axioms.   [ This chapter comes much later in the story.  ]

Alright back to the inspiration for this scaling/repeated addition multiplication thinking.  The book!

The Magic

Because I read this book to my kindergarten daughter and second grade son (neither of which have formally studied multiplication in school), the magic described is in that context.

This book provided multiple opportunities for them to reason about the effect of Rumpelstiltskin’s multiplication stick. At each page turn, my son requested I pause for him to figure out the answer to each scaling problem (using repeated addition). Then they both would count and check the answer against the illustration. After Rumpelstiltskin multiplied four branches in the fireplace by ten and my son shouted forty!, “Oh my gosh. How did you do that?! Siena wondered. Magic (to her).

They enjoyed pointing out the multiplication sign (×) on the stick and finding it on each page. And of course, they loved when Peter makes Rumpelstiltskin vanish with a “Rumpelstiltskin times zero!”

Calvert’s story  The Multiplying Menace illustrates that there there is more to the story of multiplication than repeated addition.   And while they may not know the whole story, yet.  It is important that we continue to share mathematically honest stories about multiplication in the meantime.


Have a magical math book you’d like share? Please go to the Shared booklist to find out how.  If you’d like to receive these magical math book posts each Monday, be sure to follow this blog in the side bar of this page.

Thanks and see you next Monday! #mathbookmagic

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