## Two New Prime Factorization Blankets

I got two new great-nephews in 2022!

They are both second sons of their families. I didn’t knit their older brothers’ new blankets because I had barely finished my nephew Martin’s blanket, and they came so close together, I settled for two little cardigans.

But for Kellen and Tobi, I decided it didn’t matter if I was late finishing (and I was) — they needed Prime Factorization Blankets!

The idea is the same as my first Prime Factorization Blanket for Arianna:

Rows of Entrelac squares (or diamonds), going from 1 to 99. 1 is white and I put rows of white squares in between the rows with other numbers. After 1, every prime number gets its own color. For composite numbers, I put sections of the colors for each factor. So 4 gets two sections of 2, 6 gets a section of 2 and a section of 3, and so on, all the way up to 99, which gets two sections of the color for 3 and one section of the color for 11.

Kellen is modeling his blanket in the picture above, and here are some more pictures of it.

First the blanket as a whole. I knew he was a boy, so I used lots of blues, with 2 being yellow.

The corner at the start with the missing square for zero:

And the right bottom corner with some numbers labeled:

I don’t think I knew Tobi’s gender when I started his blanket, and I decided to try for bright colors instead of pastels, so 2 was red. Here’s the whole blanket:

Detail for the lowest numbers:

Detail for the highest numbers:

And some primes at the top of the blanket:

So much fun! (Tobi’s parents, if you read this, I need more pictures of Tobi modeling his blanket!)

And yes, I’m happy to report that my youngest sister is now expecting a baby, and he’s going to get a prime factorization blanket, too! I learned tonight that he’s a boy, so 23 is going to be blue.

Babies and math are beautiful!

## Review of Friends Beyond Measure, by Lalena Fisher

Friends Beyond Measure

A Story Told with Infographics

by Lalena Fisher

Harper, 2023. 40 pages.
Review written March 17, 2023, from a library book.

Friends Beyond Measure is a simple picture book story – illustrated with charts and graphs.

It begins with a Venn diagram on a spread that says, “It started the day we met.” We see the two girls meeting at a carnival and a big Venn diagram on the opposite page shows lots of things about the girls, with many things in the intersection of common interests. We see right away that Ana (“Me”) likes charting.

And all the rest of the charts are shown as Ana’s doing. I think my favorite is the line graph charting volume (of a party) vs. comfort, and we can see that Ana is much happier with loud parties than Harwin is. But they decide together to stay for cake and then leave.

However, things get sadder when the girls learn that Harwin is going to move across the ocean to England. Then instead of charts of fun things they’ve done together, we see charts about how few days left and a flow chart of what would happen if Ana tried to sneak in Harwin’s suitcase and the mixed emotions of trying to be happy for her friend but sad for herself.

The final chart is a map of things they’re going to do together in the future, all over the world.

It all adds up to a fun book with a sweet friendship story — and lots of different kinds of charts. There’s an explanation of the different kinds at the back. I think that there’s a certain kind of kid who may just take off making their own charts after reading this book. For the rest, it does expose them to different ways of presenting information.

Find this review on Sonderbooks at: www.sonderbooks.com/Picture_Books/friends_beyond_measure.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Disclaimer: I am a professional librarian, but the views expressed are solely my own, and in no way represent the official views of my employer or of any committee or group of which I am part.

What did you think of this book?

## Review of Some of These Are Snails, by Carter Higgins

Some of These Are Snails

by Carter Higgins

Chronicle Books, 2023. 52 pages.
Review written June 12, 2023, from a library book.
Starred Review

Some of These Are Snails is a bright and colorful book of graphic design on a white background introducing the early math concepts of classification and quantity.

The pictures are made up of simple shapes – mostly circles, squares, and triangles. They’re distinct colors. They’re in different sizes. Some have enough detail added to turn them into animals. And many have spots or stripes.

Starting simple with “turtle is a circle,” several things are introduced, and it quickly builds in complexity. I particularly like this page:

purple circles
small circles
circles in a square [Above those words, nine small circles are arranged in a square formation.]

And then the book starts asking questions:

can you sort by color?
can you sort by size?
can you sort by shape or find the animals with eyes?

As it progresses from there, you see many things on a page, and the book begins using vocabulary like “all,” “some,” “each,” “a lot.”

And then some more questions:

what is one?
what is some?
where is all and
where is none?

who’s stripiest?
who’s spottiest?
who’s wiggly wigglier wiggliest?

The book finishes with more images and more rhymes about them (wonderful rhymes that you want to read aloud), still looking at the pictures by color, shape, size, and animal.

As with most picture books, you really do need to check this out and hold it in your hands, preferably with a child in your lap. This book is playful and beautiful and best of all, encourages small children to talk about what they know about the simplest kinds of math.

Find this review on Sonderbooks at: www.sonderbooks.com/Picture_Books/some_of_these_are_snails.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Disclaimer: I am a professional librarian, but the views expressed are solely my own, and in no way represent the official views of my employer or of any committee or group of which I am part.

What did you think of this book?

## Review of Edward Lorenz and the Chaotic Butterflies, by Robert Black

Edward Lorenz and the Chaotic Butterflies

by Robert Black

Royal Fireworks Press, 2022. 127 pages.
Review written January 8, 2022, from my own copy.

Edward Lorenz and the Chaotic Butterflies is a short but thorough biography of one of the founders of Chaos Theory.

Edward Lorenz got interested in meteorology because that happened to be where the U.S. War Department could use his mathematical skills when World War I started.

The book explains how the science of meteorology was developing as computers were developing. And when they tried to model the math of weather forecasting, it was so complex that those two things went together. In fact, because Edward Lorenz had a desk-sized computer in his office at M.I.T., he was able to notice things that other researchers had a harder time studying.

They talk about his initial discovery. He wanted the computer to repeat some calculations but go farther, so he started by typing in the results from already-calculated numbers. But the results the second time through were completely different. He realized that was due to a rounding error — he hadn’t printed out all decimal places of the solutions, so he was actually starting with slightly different numbers.

But why did slightly different starting numbers make a huge difference in results?

I like the way the book describes the equations he used as both unpredictable and stable. The equations are relatively simple, but the results vary wildly. The book even shows how you can do the same thing with a home computer (much smaller than a desk) and an Excel spreadsheet.

I did gloss over some of the equations, but I got the idea of how it all works, and I think students can do the same as me or dive in deeper if they want to know more.

A quick biography of a notable mathematician who started a whole new field of study and showed that not all of reality is linear and predictable.

rfwp.com

Find this review on Sonderbooks at: www.sonderbooks.com/Teen_Nonfiction/edward_lorenz.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Disclaimer: I am a professional librarian, but the views expressed are solely my own, and in no way represent the official views of my employer or of any committee or group of which I am part.

What did you think of this book?

## Review of How to Count to ONE, written by Caspar Salmon, illustrated by Matt Hunt

How to Count to ONE

(And Don’t Even THINK about Bigger Numbers!)

written by Caspar Salmon
illustrated by Matt Hunt

Nosy Crow, 2023. First published in the United Kingdom in 2022. 32 pages.
Review written July 11, 2023, from a library book.
Starred Review

How to Count to ONE is one of those interactive picture books that speaks directly to the child reader, and this one is all about subverting expectations of counting books.

It starts with a picture of an apple, and asks the reader to count it.

Then it says, “Now for something bigger . . .” and gives them an elephant to count!

Next you think, “Ah, here’s more to count!” because the spread is filled with two giant whales. But instead, the narrator asks:

How many SAUSAGES do you see?

[There’s one, floating on top of the spout of a whale.]

And that’s how things go, with pictures of more and more things — but at least one object in the picture there’s only one of — and that’s what the reader is asked to count.

It’s amazing how difficult it is to only count the one thing. And the narration plays off that. Here’s one example spread:

So, here we have . . . some rhinos,
a few baboons, a number of snakes,
several ants and butterflies,
and ONE giraffe.

Using your counting skills, please count the giraffe.

I hope you didn’t count the other animals.
Remember, this book is about counting to ONE!

Finally, the narrator accidentally asks the reader to count the goldfish, instead of the goldfish that is wearing glasses — leading the reader to say “Two.” See, even the narrator makes mistakes!

But it all ends with the narrator thinking maybe you’re better at counting than they thought, so the reader is presented with one prize to count.

And if they’re just dying to count higher by this time, the endpapers show one hundred things to count.

I love about this book that some children won’t be able to resist counting things and other children will start looking to spot what there is one of. And it’s all in a playful package for plenty of laughter — while counting.

nosycrow.com

Find this review on Sonderbooks at: www.sonderbooks.com/Picture_Books/how_to_count_to_one.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Disclaimer: I am a professional librarian, but the views expressed are solely my own, and in no way represent the official views of my employer or of any committee or group of which I am part.

What did you think of this book?

## X + Y, by Eugenia Cheng

X + Y

A Mathematician’s Manifesto for Rethinking Gender

by Eugenia Cheng

Basic Books, 2020. 272 pages.
Review written May 9, 2023, from a library book.
Starred Review

I love this book so very much! And I’m so glad I finally got it read — had meant to ever since it came out. I’m going on vacation this week and decided to finish it before I left — and Wow! — it’s full of transformative concepts.

Now, if you’re not a math-lover like me, please don’t be put off by the fact that this book is written by a mathematician. She works with Category Theory, a branch of mathematics that doesn’t necessarily deal with numbers, but about relationships and how things interact with other things.

In this book, she uses category theory to take on gender. And wow! What a lovely job she does.

Now, she starts with something I related to completely — the lack of women in higher mathematics. I started a PhD program in math at UCLA immediately after college. I was one of only 5 women out of 120 new math graduate students that year. Three of the other women were in master’s programs, not doctoral — and I ended up dropping out a year and a quarter later with my Master’s. So I am very aware that there’s a gender disparity in mathematics.

The first part of the book looks at difficulties with even talking about gender disparities. She looks at problematic arguments about nature or nurture.

And then she takes us to another dimension. She takes the gendered dimension out of the discussion and defines some new terms.

We need new, ungendered language in order to separate character traits from gender and have less divisive conversations in which people don’t have to get defensive about “not all men” or “not all women” being a certain way. Because indeed men are not the same, and neither are women. Not all men are aggressive, competitive, risk-taking, and unempathetic. And even those who do behave in those ways might only do so because of social pressure and the idea, perpetuated by social norms, that this is how to be successful in society. When certain behavior is rewarded by society many people will strive to behave in those ways even if at some deeper level it makes them unhappy.

The ungendered language she comes up with are the terms “ingressive” and “congressive.” Ingressive behavior focuses on the individual, and congressive behavior focuses on the group. Here are her definitions:

ingressive: focusing on oneself over society and community, imposing on people more than taking others into account, emphasizing independence and individualism, more competitive and adversarial than collaborative, tending toward selective or single-track thought processes

congressive: focusing on society and community over self, taking others into account more than imposing on them, emphasizing interdependence and interconnectedness, more collaborative and cooperative than competitive, tending toward circumspect thought processes.

This is not a clean dichotomy even though some aspects sound like exact opposites of each other. It’s not a classification of people into two camps. It’s a way to evaluate behavior in a flexible and dynamic way to reflect the fact that people are flexible and dynamic day by day and also over the course of their lives, not fixed and rigid.

She goes on to talk about spaces that are more ingressive and spaces that are more congressive. Interesting to me, she left her position at a university and began a career teaching math at an art school. Yes, she found many more women at the art school, but she can go beyond gender and talk about how it’s much more set up as an ingressive space. And she’s happier working in that space.

She describes examples of other people, both women and men, who gave up more ingressive careers (after achieving success) for more congressive roles.

And she talks about how math education is set up to reward ingressive behavior, with lots of testing and competition. This was interesting to me, because at first I thought, oh, I must be mostly ingressive because I love competition, and really enjoyed math competitions when I was a student.

But then I think about my happy switch from teaching college math to becoming a librarian. I’ve said many times, “In the library, I still get to help people learn, but now I’m on their side! As an instructor, I had to test them, and I felt like their adversary. Now I get to show kids the fun side of math.” So I have to stand as another example of someone who turned out to be much, much happier in a more ingressive career path.

I was also fascinated by her ideas to make education – yes, even mathematics education — more congressive. She suspects that could attract more people to the field who might be turned off by the current more ingressive way it tends to be taught. I loved when she started elaborating on that idea:

Congressive math is what I have been teaching to art students for several years. I think it’s also what is introduced to children at the very beginning of school, when it’s all about play and exploration with blocks and toys and other things they can touch and feel. It comes back around to being like this at a research level, but by that time I think we’ve already put off far too many congressive people with the phase in between.

I think math should be congressive all the way through. We really don’t need to train people to be human calculators anymore, now that we have actual calculators with us more or less all the time (for example, on our phones). So math could be more congressive by being about exploration and processes. It could be more about ways of thinking than about knowledge.

I would like to see a non-cumulative curriculum so that each stage doesn’t depend on the previous stage. The traditional model is more like a series of hurdles that get higher and higher and are specially designed to weed people out at each level. Not only is this ingressive, it’s also counterproductive, as we are not weeding out the right people.

But I shouldn’t go on so much about math, because that was a side point. The really beautiful thing about this book was how it took gendered language out of discussions about our society. I suspect you could also use these ideas in discussions about race.

I believe that the new dimension of ingressive and congressive traits can help us overcome bias that can’t be addressed on the dimension of gender alone. I think this is how we can deal with implicit bias in the system that comes from our association of character with gender, and how we can deal with indirect bias that comes from favoring ingressive behavior. But I think it will also have an effect on explicit bias, as it is a way for everyone to escape one-dimensional gendered thinking in their heads and think more clearly about what contributions to society we want to see. With every individual who escapes that thinking, the hold of both implicit and explicit gender bias will be lessened.

And the book finishes up with a vision of what things could be like in a society that encourages congressive behavior — and offers tips on ways to be more congressive in your personal life and perhaps in your personal environment (such as her math classrooms in art school). There’s even a helpful appendix at the back that demonstrates how you can respond to personal attacks with congressive statements instead of ingressively attacking back.

Now I’ll admit, I was predisposed to love this book having been a woman working in the field of mathematics, even if it was at a lower level than Dr. Cheng. But I’m also delighted with and impressed by her ideas. May we work together, in our lives and in our communities, to make the world a more congressive place. I highly recommend this book, and would love to discuss it with others who have read it.

Find this review on Sonderbooks at: www.sonderbooks.com/Nonfiction/x+y.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Disclaimer: I am a professional librarian, but the views expressed are solely my own, and in no way represent the official views of my employer or of any committee or group of which I am part.

What did you think of this book?

## Review of Too Many Pigs and One Big Bad Wolf, by Davide Cali and Marianna Balducci

Too Many Pigs and One Big Bad Wolf

by Davide Cali and Marianna Balducci

Tundra Books, 2022. 32 pages.
Review written January 8, 2023, from my own copy.
Starred Review
2023 Mathical Book Prize Honor Book, Grades K-2
2022 Sonderbooks Standout: #9 Silly Fun Picture Books

This very silly book is a counting book that’s not really a counting book.

Here’s how it begins:

Once upon a time, there were three little pigs.
Then the wolf ate them.
THE END.

This story is too short!
I want a longer one!

In the longer story on the next page, there are four little pigs that get eaten.

And so it progresses, the narrator adding wild things to the story, the “reader” complaining, and the result always the same.

The pigs are drawn from beads on an abacus. There are not more than ten beads on a row, and often they’re grouped by fives, so counting is easy.

The graphics, the silly stories, and the dialogue between the narrator and the objector are simply loads of fun. We never see the pigs get eaten, and there’s a feeling that these are actors that are not really harmed, since the same abacus gets reused. So it keeps things light and silly, despite so very many pigs supposedly getting eaten.

This is another one you’ll enjoy most if you check it out for yourself, as my description can’t do justice to how much fun it is. You can throw in some counting when you read it to your kid, but I don’t think they’ll think of it as a counting book.

Find this review on Sonderbooks at: www.sonderbooks.com/Picture_Books/too_many_pigs.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Disclaimer: I am a professional librarian, but the views expressed are solely my own, and in no way represent the official views of my employer or of any committee or group of which I am part.

What did you think of this book?

## Review of Zero Zebras, by Bruce Goldstone, illustrated by Julien Chung

Zero Zebras

A Counting Book about What’s Not There

by Bruce Goldstone
illustrated by Julien Chung

Orchard Books (Scholastic), 2022. 36 pages.
Review written January 8, 2023, from my own copy.
Starred Review
2022 Sonderbooks Stand-out: #6 Silly Fun Picture Books

I love this book in so many ways! In fact, many more than zero ways!

Here’s how the book begins:

I see one wallaby . . .
. . . and zero zebras.

Two tuna splish
and splash
and splosh . . .
. . . with zero zebras.

You get the idea!

As things progress with various jazzy animals, there’s wordplay and visual play:

Ten tigers tiptoe —
that’s how many.

There aren’t any.

Eleven llamas
like to spit.

It’s zero zebras
that they hit.

But the fun really begins after we pass twelve turtles on the page.

What’s next? What’s here?
What do you see
perching in this tree?
Why, look at that!
By now you’ve guessed.
Zero zebras — obviously.

But that’s not all that isn’t here!
Do you see zero eagles?
You’ll find them next to zero pigs
and zero barking beagles.

Then we’ve got two more spreads with rhymes about all the things pictured that they are zero of.

The finishing thought is this:

So when you want to count a lot,
don’t count what’s there. Count what’s not.

Try counting zeroes with your friends.
The list of zeroes never ends!

The final pages have thoughts from the author about zero and infinity.

So there you have it. A delightfully silly picture book that invites play and imagination and all kinds of fun — while getting kids thinking about the important mathematical concept of zero.

I did quote a lot of this book, but please let that invite you to see it for yourself, because a picture book is always best with the words and pictures together.

Find this review on Sonderbooks at: www.sonderbooks.com/Picture_Books/zero_zebras.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Disclaimer: I am a professional librarian, but the views expressed are solely my own, and in no way represent the official views of my employer or of any committee or group of which I am part.

What did you think of this book?

## Review of Bake Infinite Pie with X + Y, by Eugenia Cheng, illustrated by Amber Ren

Bake Infinite Pie with X + Y

by Eugenia Cheng
illustrated by Amber Ren

Little, Brown and Company, 2022. 32 pages.
Review written July 13, 2022, from a library book
Starred Review

My readers won’t be surprised that every time mathematician Eugenia Cheng writes a picture book, it delights my heart. I’ll list this under Children’s Nonfiction, because although it is a story, the emphasis is on the ideas.

This one tells of two kids, named X and Y, who are dreaming of infinite pie — X, pie that is infinitely wide, and Y, pie that is infinitely tall. They think of course that such pie isn’t real, but they ask their Aunt Z, who can create amazing things with her brain.

The book that follows explores infinity in many different ways, and all of them involve pie!

There are infinitely different ways you can make pie, and once it’s done, if you keep eating half your pie, it will last until infinity.

You can make a pie with infinite corners, cut pie crust infinite ways, and even make pastry with infinite layers.

I hope that gives you the idea of ways to explore infinity with pie — it’s all presented in a family setting with a fun aunt bending kids’ minds with tasty treats.

And there’s even a recipe for pie at the back! (After a spread that lays out mathematical ideas presented.) The recipe is for Banana Butterscotch Pie — and believe it or not, I couldn’t resist trying it out. (I hadn’t made a pie with crust in decades.) The pie was indeed delicious, but alas – the instructions didn’t specify how big your pie pan should be. I used a 9-inch one, and my pie was more of a tart — the filling only went about halfway up the pie crust. I think an 8-inch pie pan would do nicely. And it still tasted wonderful.

Find this review on Sonderbooks at www.sonderbooks.com/Childrens_Nonfiction/bake_infinite_pie.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Disclaimer: I am a professional librarian, but the views expressed are solely my own, and in no way represent the official views of my employer or of any committee or group of which I am part.

What did you think of this book?

## Review of Look, Grandma! Ni, Elisi!, by Art Coulson, illustrated by Madelyn Goodnight

Look, Grandma!
Ni, Elisi!

by Art Coulson
illustrated by Madelyn Goodnight

Storytelling Math (Charlesbridge), 2021. 32 pages.
Review written December 29, 2021, from a library book
Starred Review

Here’s another book from the amazingly good Storytelling Math series. All of them present math concepts in a real-life setting that will appeal to children. All of them also present cultural information, not presented as “other” or “exotic” or “different,” but from the perspective of a child within that culture, excited and proud and enjoying things their family does.

In this book, Bo is working on large colorful stone marbles for the Cherokee National Holiday coming up. The marbles are used in the game Cherokee marbles, digadayosdi, and Bo wants to sell them in his family’s booth at the festival.

Bo has a lot of marbles, and he wants to display them at the booth. But when he finds a nice tray to use to display them, Grandma tells him that the tray is too big. Their booth is small and he can display the marbles in the booth, but whatever he uses needs to fit on a small mat she shows him.

So it’s a volume problem. Bo is trying to find a container with a base as small as the mat that will still display the marbles well. And not so tall that it’s hard to reach inside.

After Bo finds the perfect container (which takes lots of tries), they show him happily displaying them in his family’s booth — and then playing Cherokee marbles together to get a break.

The book weaves in some Cherokee words, and there’s a glossary at the back along with the feature at the back of every Storytelling Math book called “Exploring the Math.” In this book, that section gives ideas of activities to help kids explore volume and area. I love the way these stories are jumping-off places for more learning.