Book Celebration

To celebrate the release of the newest great and greatest new children's math book... by which I mean Which One Doesn't Belong? by the #MTBoS' own Christopher Danielson, of course... I thought I'd recap some of my favorite math picture books. This list also has MTBoS support, as I solicited suggestions from the MTBoS for a colleague.

Get yours here.

The request was for a parent with a mathematically curious child (really, could be anyone then, am I right?) of 4 or 5 years.

Top:

• Moebius Noodles, Maria Droujkova's great book about big math ideas to explore. There were articles about calculus in kindergarten when it first came out.
• Great new book: Which One Doesn¹t Belong. OK, I'll say more. I love this book because it's clever and pretty, but also because it can teach you how to read mathematically rich literature.
• Math Curse, Lane and Scieska: just the best math book ever written. Nearly anything can be a problem, you know.

Great:

• Anno's Mysterious Multiplying Jar, or anything by Mitsumasa Anno. Just charming books, and lovely besides.
• Spaghetti and Meatballs For All, Marilyn Burns: my favorite of the explicitly mathematical genre. Tang and Murphy have their place but Burns is the queen of the genre. (Greedy Triangle, Smarty Pants, $1 Word...) 
• Princess of the genre, Elinor Pinczes: One Hundred Hungry Ants, A Remainder of One, ...
• Infinity and Me by Kate Hosford
• Tessallation!  by Emily Grosvenor
• Grandfather Tang’s Story, by Ann Tompert
• The Dot and the Line, by Norton Juster

Biography:

• Paul Erdos, The Boy Who Loved Math by Deborah Helligman
• Fibonacci, Blockhead by Joseph D'Agnese
• Eratosthanes, The Librarian Who Measured the Earth by Katherine Lasky
• Einstein usually gets lumped in here. Not a mathematician, technically, but if you're being that technical, loosen up! Try Odd Boy Out or On a Beam of Light

What to do:

• Chris Hunter has a bunch of recommendations plus how to use them.
• Another post with a lot of suggestions at Algebra's Friend
• Suzanne Alejandre suggested the old (aka classic) NCTM guide, Welchman-Tischler's book, might provide ideas? (She found a digital copy; use the NEXT button)
• Marilyn Burns (herself) just suggested this starting point from PBS

Possibly for older, but like Madeline L'Engle, I think people underestimate kids:

• The Phantom Tollbooth, by Norman Juster
• The Man Who Counted, by Malba Tahan
• Flatland, by Edwin Abbott 
• The Number Devil, by Hans Magnus Enzensberger
• The Adventures of Penrose the Mathematical Cat, by Theoni Pappas
• The Cat in Numberland, by Ivar Ekeland and John O'Brien
• A Wrinkle in Time, Madeline L'Engle (First I heard of a tesseract.) There's an audiobook where L'Engle reads it herself. Highly recommended.

And please add your own suggestions!

PS:

• Cindy Whitehead saw that I missed the Sir Cumference books, by Cindy Neuschwander, and suggested the Go Figure books, by Johnny Ball.

Six Sides to Every Story

I finally got a chance to teach the Hexagons.

Christopher Danielson has a distaste for boring old quadrilaterals, so he came up with teaching hexagons. (Blog posts: one, two and three.) Dylan Kane wrote briefly about his experiences with them at NCTM:NOLA, and Bredeen Pickford-Murray has a three post series on them.

For me, the idea of type in geometry is pretty complicated. You need to have a large variety of the thing that will have types, you need experience with that variety, you need to be doing something with them - often describing them. (Boy, you're needy.) Types are the way we sort, which means attention to properties and attributes. Definitions come after types, as the more we work with something the more we need precision. Making a good definition is a great task for forcing you to realize some of your assumptions about the objects. 

With a group of preservice elementary teachers, quadrilaterals offer a lot of these opportunities.   Mostly they are at the visual geometry level, and debating whether a square is a rectangle is a good discussion. (Most have been told that, but don't really believe it.) This class I'm teaching is all math majors who have had a proof writing course and many of whom have had a college geometry course. Their quadrilateral training is not complete, but they are strong in the fours.

Sorry.

For me this made the conditions right for trying hexagons. I was up front with the idea that part of this was to put them in their learner's shoes, about trying to make sense of this material seeing it for the first time. So we started with variety and description. 

The first day we played guess my shape on GeoBoards. One person describes and another or a group tries to make what they're describing. For more challenge, the describer can't see the guesser's geoboard either.  I limited us to hexagons. I also tried to bar instructions; you weren't trying to tell them how to make it, but rather describe it so well they could make it.  

After playing in their groups, we tried one person describing while all of us tried to make it. When she was done, several people presented their guess. That gave us a sense of the holes in her description. But on the reveal, there was appreciation for what she was getting at, and several good suggestions about more description that would have helped. Then we looked for what made descriptions helpful.

Brainstorming

With that experience, we brainstormed some possible types. The homework was to draw two examples of each possible type, and come up with reasons that is should or shouldn't be a type. Also they read Rubinstein and Crain's old MT article (93!) about teaching the quadrilateral hierarchy. 

The next day I gave them some time to make classes of hexagons within their table, with the idea that we'd come together to decide on class types. We went totally democratic: each group would pitch a type to the class, and then we voted yea or nay. The one type we started with was a regular hexagon. Mostly I was quiet in the discussions, and I abstained from the votes unless I felt strongly about it. Once we had a pretty good list of types, we did class suggestions for names. I shared how historically names are often either for properties or what they look like. The table that proposed the shape had veto authority over the name. But Channing Tatum almost became a type. Several people reflected that this was a pretty powerful and engaging experience.
















Homework was to make a set of 7 hexagons. At least one is exactly three types, three are at least two types, two are exactly one type and one of your choice.

The third day we opened with one of my favorite activities: Circle the Polygons.






We do rounds of finding out how many polygons people or groups have circled, and then they can ask about one of them. Then they recallibrate. If students have no experience with polygons (2nd or 3rd grade) I will start with some examples and non-examples. Once we're agreed with what a mathematician would say, I asked them to define 'polygon.' Closed and straight edges are pretty quick, and someone usually thinks about 2-dimensional. (Although I usually wonder if they have a counter example for that in mind.) The last quality is a struggle. After we have a couple student descriptions, I share that mathematicians also had trouble expressing non-intersecting, no overlap, no shared sides and just made a new word: simple. Then I asked a spur of the moment question about what did they see as the purpose of definitions and, wow, did they have great ideas.

So we worked on definitions of the hexagon types, sorted them, and looked at each others' sets. Super nice variety. People brought up a few that were tough to type, and they caused nice discussions on which properties were possible together and some good informal arguments on connections. The note to the side was a discussion of Dan Meyer's "If this is the aspirin, what's the headache?" question for definitions. We agreed that most lessons give the aspirin well in advance of any headache they might prevent.

To sum up the experience, I asked them what they noticed about what they had done, especially in connection with the Common Core standards for shape, and what advice they would give to teachers who were asked to teach these things.

To finish up, we did one drawing Venn problem together. Come up with labels, and draw a shape in each region that fits or give a reason why you can't.

So thanks to Christopher and this great group of students for a great three days of mathematical work.


P.S. Student reaction: several of these preservice teachers chose to write about the hexagon experience for their second blogpost. If you want to hear about it from their perspective...

• Jenny thinks "It is time for teachers to get away from the cute activities that are fun for the students and get into the real meat of teaching these shapes." 
• Chelsea is thinking about adapting it to quadrilaterals. 
• Heather wrote about the inquiry aspect.

My Oreo Lesson

Finally... my chance to do the oreo lesson!

I'm teaching one of our math for elementary education courses and the content includes measurement and statistics. I love measurement as a context which needs statistical understanding. Measurement introduces variability, and has a strong need for producing a number to represent typical. If the question is, "How tall is the ceiling?" then 2.60, 2.7, 2.725, 2.73, 2.735, 2.735, 2.74, or 2.76 meters is not a satisfying answer.

The oreo lesson, if you are unimaginably unfamiliar with it, is the brainchild of Christopher Danielson, aka @Trianglemancsd, the purveyor of much fine snack food mathematics. (All the oreo posts; this one is sort of a wrap up.)

Previous to this lesson, we investigated measurement, did an introduction to statistical typicals, and worked on statistical displays. (Two of those covered in a previous blogpost.) On the day before oreo day, I brought three packages of oreos to class: regular, double stuf, and mega stuf. Their interest was definitely piqued; it was like they could smell the sugar. Not much mathematical interest, though. So I prompted - what might a mathematician wonder about this? They immediately jumped to the idea of is it really double, and what is mega. Then we brainstormed together - what do we need to gather data on for the next day?

Their list:

OREO: data to collect

weight of the whole cookie

weight of white stuf in each cookie

height of each cookie (mm)

diameter of each cookie

weight of cookie/black sides

height of black cookie

height of white stuf

diameter of white stuf

how many of each size fit in a specific container/height

volume by displacement

compare deliciousness of different types

nutrition information

stuffing v serving size

calorie content (burning)

Not bad. Calorie burning turned out not to be viable with that short of a notice... but I'd like to see it! I made a data sheet, so that we'd have a whole class worth of data, and a google spreadsheet to share.


The points about measuring like a scientist (half of the smallest unit) and recording to show the accuracy measured are obviously still in progress. Also the statistical thinking of gathering and using data need more development - most were happy to answer the main question with just their measurement. "It's double." "It's more than double." "It's less than double." No one used the measurements, they went entirely by weight.

That wasn't what bothered me. I expect those kind of goals to take time.

What bothered me was that they weren't into it.

They were excited about the cookies, and figuring how many each person got, and eating the cookies afterward. But they weren't into the math.

Dave Coffey sometimes recounts (or makes fun of me for) how I want to be obsolete. Sitting back and watching students direct themselves at the end of the semester. I always want to hand off to the students. Have them make it their lesson. Look, here's a pile of data! On something interesting! What can you do with it? What else could we look into? How many ways can we come at the question?

But on this day, they said no.

My personal metaphor for this is a Smothers Brothers routine. (That's how old I am.)


(The whole brilliant bit... the show was amazing. Steve Martin got his start there as a writer, for example. They used their folk singing to make the show safe for sharp political commentary. Like we use math class as a ruse to get students problem solving and thinking critically. They were cancelled and replaced with Hee Haw.)

So this lesson felt like, "Take it, class!"
"No."

My response was to ask them to make sure they got all their group's data, and to write about the measuring and their answer to the question for a standards based grading assessment. And this is a compliant class, so they did, and did a good job on it. But that's far from the peak experience for which I was hoping with this lesson.

Part of the problem, I think, was in my desire for efficiency. By introducing the problem in the previous class and then making a record sheet, I took the initiative from them. They went into fill in the blank mode, from long habit in math class. Another part of the problem was lack of a focus, in the workshop sense. I think I should have discussed statistical thinking with them, and how that's different from single measurement thinking. It's all about the data! This is very reminiscent of the Barbie Bust. It was my problem and my lesson. "My" doesn't help me be a better teacher. (Gollum.)

Reflecting afterward, I think my high expectations helped create the sense of disappointment, like an overhyped movie.  And it led me to rush into a lesson instead of building suspense and anticipation.  I think this kind of experience contributes to teachers who "tried that once" and that was enough to turn them off of inquiry-based learning.  In the end it is the learning that needs to be the center of engagement, not the cookie.