2012: An Amazing Year

I've been looking back on a year's worth of posts and it is amazing to me how much math my daughter and I have done, both individually and collectively.

We've gone on math walks, grown our math eyes, read books, talked math, played math, made our own math games and, especially, built our math understanding one piece of chalk, pipe cleaner, straw, woven paper strip, camera snap, colored pencil, bead, craft stick, scissor, clover, flower petal and glue stick at a time.  Here are the top posts for 2012, but for every one in this list there are many, many more just as important to me (even if less read):

1. Solids and an Observation on Scale

2. Beading Attributes: Pattern, Color Shape, Size and...Straws!

3. Is it Cheating to Use the Multiplication Chart?

4. New Math Game: Factor Dominoes!

5. Math By Design: Paper Patterns

6. And the Award for Favorite Platonic Solid Goes to...

7. Weaving Geometric African Motifs: Part 2

8. A Game of Her Own: Discovering Division

9. Seeing Stars

10. Math in Action: Catapults!

And, an excerpt from a post that is among my personal favorites from the year, The Elephant in the Room:

This flurry of math in the making has me reflecting on how I used to think that math was some kind of  inaccessible, abstract magic trick, a sort of in-joke that excluded us common folk, but now I realize that math is completely not that at all.  The reality of math as most of us know it is like that story where three men are standing in a dark room touching different parts of an elephant.  None of them has the full picture because they're only perceiving individual elements of the whole animal.  

The reality, I'm discovering, is that math is just like that elephant: a large, expansive, three dimensional, intelligent, sensitive, expressive creature.  The problem is that most of us have been standing around in that dark room since about kindergarten, grasping its tail, thinking "this is what math is and, personally, I don't think it's for me".  We've been blind to the larger, incredibly beautiful picture that would emerge if only we would turn on the lights and open our eyes. 

Thank you, everyone, for reading, and huge thanks to the many readers and supporters who helped make 2012 an incredibly fantastic year of mathematical discovery and growth.

Circle Discoveries

I went to the Y.  I came back.

"Look, Mama, I've been discovering things about circles while you were gone."

"Oh really?  What did you discover?"

(It looks like she discovered an old pad of graph paper and was exploring a drawing of concentric circles made by me a while back - she drew in the divisions -- and then moved on to make one of her own with a compass.)

"Well, I discovered that there are lots of ways to divide a circle...and I discovered that no matter how big or small a circle is you can still divide it the same way."

 "Well, that's very interesting!  But what's this?"

"This is a cylinder and these are circles."

"Cool!  And what's this one?"

"These are ovals.  They're squooshed (sic) circles.  This one has one triangle with a curve and the others are rectangles."

"Hmmm...I don't think I agree with that.  Remember that a rectangle has to have four right angles.  That's why they call it a rec-tANGLE..."

"Oh...yeah."

I really think that using geometry and shapes to explore number concepts is a strong combination.  I've been working on a new idea that's sort of algebraic, geometric and numeric all at once.  Since the kid seems to be experiencing a post-holiday renaissance I might just be able to finish up the activity with her some time soon.  Stay tuned!

A Very Math-y Holiday

It's been a wonderful holiday for us so far, capped off yesterday with 12" of snow. Besides our winter wonderland, another part of the wonderful-ness has been an extraordinary confluence of math gifts.

One of my very favorites was a scarf my eighteen year old niece knitted for my seven-year-old daughter.  Can you see the math?  It's a Moebius scarf!!!  I love the idea of having infinity around my neck and hope my niece remembers that I have a birthday coming up...

Another of my favorite math-y gifts was something unintentionally so.  These are some doll house pots and pans and table settings I gave my daughter.  As we were eating breakfast we arranged them and that's when I noticed something.. 

 "Look!" I said, "each pot has one lid and there are four pots in all.  How many pieces all together?  [I modeled skip counting by 2s]  And the table settings!  Each set has four pieces.  How many pieces all together?"

Despite a facility with skip counting and good grasp of groupings and even certain multiplication facts, the child was still skeptical.  She insisted (and has been insisting) on checking my work by counting by 1s.  I still thought this was an excellent present for math, though, especially because it highlighted her thinking so clearly.  Good thing I have Amanda Bean's Amazing Dream on the bookshelf (you know, the girl who is great at counting and doesn't want to learn multiplication until things get too numerous and hard to count by 1s).

Another great gift was one from my my mother -- a wonderful little felt board activity in perfect condition that may have been thrifted -- no name, no box.  Anyhow, in included right/isosceles triangles made out of stiff felt and a TON of pattern cards with both geometric and representational picture designs. 

I pulled it out the day after we beat the blizzard home from Ohio as a sort of after-Christmas gift.  The girl started to do a card, pictured below, and, again, it was very interesting to observe her thinking, especially how she went about putting it together. My eye goes immediately to the center and I would have built outward; she started with the top left corner and built across.

Eventually though, the tired girl eschewed all patterns and decided to do her own thing and I worked on my own card.  

Here are my favorite pattern cards so far.  The top row is a nice study of growing shapes (algebra!), the middle and bottom rows are just totally cool (the yellow mixes in with the pink -- use your light orange shapes!) and look at all that negative space!

Here's some math we gave instead of received.  I've already written about these mathematical star ornaments, but the reason we made them was to give as presents to all our friends.  The ones below are mine, but my daughter made over 40 of her own (in a different color theme).  I also made our holiday cards out little paper squares folded and then transformed by scissors as detailed in a post from the summer called Transformation.

The Kaleidograph Toy (Flora and Crystal) were supposed to be mine, but who am I to argue if the kid wants to use them too?!

And then there's the moment all the math stars aligned just right and the book on Roman mosaics arrived just when I needed it. 

All told, a very math-y holiday, indeed!  And, here are best wishes for a very Happy & Math-y New Year, a little early, to all my readers!

When the Math Stars Align: Geometric Tiling Edition

I've been experiencing some serious math kismet recently.  I rediscovered a thrifted book on 16th century mosques that I bought to read with my daughter.  We got part way through and I remembered what I was really interested in were Islamic geometric tiling designs, which spurred me to the library where I found an awesome David Macaulay book about the construction of a fictional 16th century mosque.  But still no leads on tiling designs.

A few days passed.  Today in the mail, totally unexpected, was this Christmas gift:

Not Islamic, but still fascinating and exactly the kind of resource I was looking for.  From the book:

"These geometric patterns radiate symmetry and order.  Drawing the patterns is not just a question of mechanically copying the work of someone else square by square, but of understanding the underlying structure.  These patterns are built up from simple elements which seem to 'grow' and develop in an almost organic or living way."  Perfect.

As I looked through it tonight I realized that some of these designs would be great inspiration for the mathematical paper weaving I've been exploring, and other designs would be perfect for the non-quilting paper patterns activity I developed this summer.

And, to top it all off, when I looked the book up I found that there are other books by the same author on the topics of geometric patterns from tile and brickwork, churches and cathedrals, patchwork quilts AND Islamic art and architecture which, it turns out, was heavily influenced by the geometric designs of the ancient Greeks and Romans.  So, really, if you think about it, the very most perfect book appeared at the very most perfect time.

I have so many new ideas I may never sleep again...

Thrifted Math

Here's something you probably don't know about me -- I've got something of a golden touch when I'm out thrifting.  Not always, but I do have a track record that is (to me) quite impressive!  See what you think about these highlights from the last year:

A down jacket, perfect fit, perfect color, just needed a wash and it was brand new again; I paid $4 and am on my second season with it still in great condition

A Leap Frog talking globe, like-new, retailing for $350; I paid $4 and the kid learned her continents and is onto countries, plus she makes up her own dances to the 'music of the world' option.

A sturdy elementary microscope with three lenses and swivel eye piece, like-new, retailing for $160; I paid $5 and, although we haven't really used it much, I just know it'll come in handy at some point.

We also get a lot of great science, social studies, history and art resource books at our library's resale bookstore.  And, just a week or so ago I found two old math games at Goodwill!  The first one is Scan, a "split second matching game" from the 1970's.  I found a comparable version it online offered for $50.  I got it for $1.99 and we've already had a ton of fun with it!!!

When I saw it on the shelf I immediately recognized it as mathematical in some way. There are four categories on each card: color (square, circle, square, circle), position (four black dots on a 4x4 grid), pattern (different combinations of x's and o's) and shape (various irregular purple polygons!). 

Our game didn't come with directions and at first I thought it the point was to match all four categories at once.  I even went so far as to try and analyze and sort the cards so that I could understand how it might work.  Here's one attempt at sorting by grid pattern: 

Ultimately, no pattern emerged and I also determined there were no 4-way matches to be made.  I'm not very good with combinatorics but even I can figure out (albeit after a bit of struggle) that you need more than 26 cards to have a match for every possible combination and permutation.  (Anyone want to figure it out?!  Just kidding.) 

I went online and found the directions and was relieved to find that you only have to match one category on the center card to a card on the table to win the round.  The box said it was for ages 9 to adult, but I decided to try with my 7.5 year old.  Initially I thought maybe I'd have to half-size the deck for the first few games so she'd have a chance to get the hang of it but for some reason I put out the full deck on our first game and, what do you know, she beat the pants off of me! 
 

I found the next game about a week after I found Scan.  It is not branded so I can only describe it with this picture:

Well, actually, as I was editing this post I noticed the company name Garlic Press in the bottom left corner.  Apparently they're still in business and have other similar two-sided 'self-check' math fact puzzles.  Essentially, you piece together this round puzzle by doing your multiplication facts...

...which, incidentally, are not in any particular order around the ring, which is a great feature.  When you've placed all the pieces you flip it over to see if you got them right.  If you did, you have a picture that makes sense.  If you didn't you have a good giggle about the parrot's beak and wing being in the wrong places and try again. 

I'm not that big on drilling math facts, but I thought it'd be interesting to try and a nice addition to our other work even at the sky high thrifting price of $4.99.  What do you know -- the puzzles are fun!  My daughter has only done the 0, 1, 2, 3, 4 and 10 times table puzzles so far but she's motivated to do about one per day.  I mean, who wouldn't want to solve your 8 times tables if you got a cute little tiger cub at the end of it?!

You use the same procedure to solve each puzzle, which makes me optimistic that she'll start developing some reasoning strategies around the skip counting she doesn't already know backward and forward -- like 11 times something is similar to 1 times something.  Or, if I do the 0, 1, 2, 3, 5 and 10 times first in any puzzle, the rest are not hard to figure out. 

The multiplication puzzles might be a little hard to make yourself, but if I knew about Scan and couldn't find the game itself I'd totally make a version myself with my kid or other students!  The Scan cards are just four quadrants with a basic concept and pattern rule for each quadrant.  Also, you need two copies of each card. to play the game.  It's essentially a quick-paced matching game for big kids;  I think upper elementary kids might really enjoy the challenge of making their own versions and variations.

Angles "In Real Life": Gym Edition

Want to hear something interesting?  At least twice a week someone arrives at my blog via one of these search terms:

"right angle in real life"
"acute angle in real life"
"vertex in real life"
"[geometric shape] in real life"

A couple more just came in as I was starting this post!  I'm never sure exactly what folks are looking for when they use those search terms, but I am always hopeful that once they get here they find what they're looking for.  I was thinking about all this when I was at the gym the other day and, as I walked around the track, I started noticing angles everywhere I looked.  I had my phone/music/camera with me as I walked so I started snapping some pictures.

As I looked closer I realized -- you can't have angles without lines.  Every time I found an angle I also found a beautiful interaction of two or more lines, with the added enhancement of a sunny day and its inverse, shadow.

The track is curved in places but the walls are straight.  This leads to some interesting intersections.  Here is an obtuse angle where two walls meet. There is also a lovely bisecting line that creates two acute angles where the straight boards meet to 'turn a corner' as it were:

All sorts of lines here!  Red lane markings run parallel to each other.  The metal what-ever-it-is runs perpendicular, creating right angles.  The wall comes into the floor at a right angle, and the light shining from the basketball court around the corner creates a lovely acute angle:

And the basketball court itself!  There are a lot of right angles here, and some lines that intersect but don't really make angles.  Why is that?

A rectangle must have right angles...

But its shadow doesn't!  Look at this lovely rhombular patch of sunlight!  Two acute and two obtuse angles of winter sun.

I didn't need a bench to sit on (my workout was basically shot at this point and my heart rate was still barely above resting) but if you do, hopefully you'll find a nice sturdy one made with right angles.

Looking out into the day, the window is divided by perpendicular lines which make right angles.  How many right angles can you find in this picture (both inside the gym and outside it)?

Behind the basketball hoop are the wires that help keep it suspended in air.  I see three intersecting lines that create an asterisk 6-star as well as six acute angles:

And more lovely shadow angles!  This is the big curtain that divides the two sides of the basketball court.  Look at the lovely obtuse angle the light creates!  

Outside the gym a plethora a lines and angles on the way to the parking lot.  

Sun and shadow get the final word. 

____________________

Malke Rosenfeld delights in creating rich environments in which children and their adults can explore, make, play, and talk math based on their own questions and inclinations. Her upcoming book, Math on the Move: Engaging Students in Whole Body Learning, will be published by Heinemann in Fall 2016.

Happy Christmas to...Me?!?

The doorbell rang.  The kid immediately jumped up from her lunch and launched herself at the front door, yelling "I'll get it!!  Maybe it's something for me!!!!"

Yes, lunch time is around when the mail is delivered at our house.

I called after the fleeting figure, "It's probably for me. It's my Christmas present."

Door opens, cute little square package on the doorstep.

"What do you mean, for you?" the girl asked, incredulous.

"Oh, you know, Grandma sent us money for presents.  I've already bought the presents she wanted you to have.  She gave me and Papa money too so we could buy presents for ourselves," I countered.

We opened the box  She looked curiously at the two little packages.

"One of them says Crystal the other says Flora, " I said. 

"Open Crystal!!" she said.

"Okay, let's see what they look like."  I opened up the Crystal pack and laid out each card one by one. 

"And, look!  There are two colors on each card, one on each side."  I took a few and started layering them.  "What if I put this one on top? And then turn it?"

"Oooohhhhh!" we both squealed.

Then I opened the Flora package.  The girl was ALL over it.  "Let me try!!!"

I purposefully did not buy Kaleidograph for my daughter.  I thought about it, but after many, many months of being called "math mommy" in an accusatory tone, I decided that these could be my special toy.  Given her enthusiastic response, though, I guess I'll let her keep using "my" present whenever she wants! 

I loved playing the Kaleidograph game online, but I wondered if the hand-held version would differ.  Both Kaleidograph sets are lovely to handle and look at, and might be fun to trace onto paper as well. I'm glad I went ahead and bought the physical version.  Even if it's not really mine anymore!!

A List, Some Links and a Thank You!

I've been doing a bit of virtual housecleaning lately.  The Math in Your Feet website is newly and thoroughly updated, and with a new look to boot!  While making the shift to the new website I was inspired to formalize an idea that's been brewing for the last year, something I'm calling Math by Design.

I've realized that all the math and making stuff I've done with my kid, and others kids as well, over the last year is really an extension of the approach I use when teaching Math in Your Feet.  Math by Design: Paper Patterns is a story of one activity I developed for my summer programs but it is also the best description and illustration of the Math in Your Feet approach being applied to a medium other than percussive dance. 

I've also put a new list of "Key Posts" up on the right hand column of this blog, but I thought I'd share them here as well.   I think my favorites are Marveling at Moving Patterns and The Elephant in the Room -- both describe, in different ways, a moment in my math journey when I realized just how connected math (structure, order, pattern) is to our daily lives. Here's the full list:

Dance & Math
Marveling at Moving Patterns
A Big Discovery (Attributes)
A Moving Classroom: "Find Your Center"
Chicken or the Egg? A Math Integration Tale
Teaching Below the Surface
What You Can Learn from a Square

Math & Making
Solids and an Observation on Scale
Beading Attributes
Scissor Stories: Tales of Transformation
Stars, Factoring & Patterns
Math by Design: Paper Patterns

The Beauty of Math
The Elephant in the Room

Finally, I just realized I missed my 2nd blogiversary (back in October) so I'll take this opportunity now to say thanks to all of you who read, subscribe, follow and/or comment in this blog!  And, perhaps the Math in Your Feet Facebook page would also be of interest to you as well?  It's a pretty fun place, if I do say so myself!

Friday Fractal Fun! (Video)

Watch the world's biggest people-made fractal being made! Time-lapse, 90-second video. Awesome!!

Mathematical Star Ornaments

I've finally succeeded in bringing my daughter (age 7.5) completely over to the mathematical stars camp thanks to this post at The Crafty Crow called "Woven Cookie Stars." 

My daughter is (in)famous for knowing her own mind and possessing very firm opinions about what interests her.  More power to her, I say, but it's required some flexibility on my part.  Over time I've been most successful in influencing her interests when I engage in my own colorful math related projects and 'just happen' to leave some scraps around for her to work with...if and when she so desires.

It was in this environment that I embarked upon our mathematical star inquiry this summer.  The details of this journey can be found in my Seeing Stars post, but I will quickly point to the picture below as evidence that although she often feigns disinterest in the 'math-y' things I do, her mind and heart are still permeable and receptive to the beauty and complexity of math, including stars.  Here's a picture from the child this summer showing her thinking about how stars are structured (explained by her in the post Plant the Star Seeds, Watch them Grow):



We really haven't done much with stars since late August.  I was stalled after a lackluster experiment with 'spinning' stars around nails; the whole thing was just too futzy.  But when I saw Crafty Crow's post I knew it was the answer I had been looking for.

I have lots of cardboard lying around.  I have scissors.  I have spice jar lids to trace.  I have embroidery floss.  My daughter saw my first efforts last night and immediately had to try it out herself.  She used up the only sparkly embroidery floss we had on hand, so today we went to the craft store and she chose her own thread -- sparkly gold and purple and some lovely shades of green. When I said I wanted to take a picture of her stars, this is how she arranged them, in all their glory:

Mathematically speaking, she's figured out a lot all on her own  Her stars are all variations of twelve-pointed stars (12 stars).  She started by wrapping thread around the circle, sort of randomly.  From there she very quickly figured out how to make her favorite star, the six pointed star she calls "the Jewish star" which is created using two overlapping equilateral triangles.  She made about twenty of those, and the last ten were completely uniform.  I encouraged her to see what would happen if she made two more triangles on top of the 6 star and lo!  There was a 12 star that echoed her work in the activity I developed for her, described in the post Stars, Factoring and Patterns.

I also encouraged her to try and make an asterisk star and wondered what it would look like if she used a combination of colors.  I really wish I had her aesthetic -- her stars are so gorgeous.

Here is what I was working on while she followed her bliss.  Initially I wanted to see how many 12 stars I could make which led me to play around with layering one color/version over another.  On the left are set of {12, 5} stars, one in each color.  The next column is a {12, 4} star combined with an 12 asterisk.  I think I like the one on top the best.  The final row (far right) is my least favorite but still pretty.  It's a {12, 3} star (see the squares?!) with a thick asterisk. (Here's more about notating stars.)

 
Every time I finished a new version or my daughter tried a new color to overlap the sparkly thread we ooooohed and ahhhhhhed.  It was a delightful morning, to say the least.  Just for fun, I experimented with a 16 star.  A little too busy for such a small space, but I do like the the one on the left.

And, although most of my time was spent trying to keep up with my daughter's demand for "more circles!" I still had time to experiment with a snowflake feel.  'Cause, you know half of twelve is six and six is a snowflake.  I want to come back to this soon.

So, by now, maybe you're itching to get started.  Remember, all the instructions are at The Crafty Crow.  And, remember, both my daughter and I got started by just playing around with the materials. Our first two or three tries each were quickly discarded but the process was incorporated into these darling beauties. Have fun!

Half A Cake

"Hey, tomorrow's your half birthday!  I know, let's make a cake to celebrate!"

Turns out the recipe called for 1 1/2 cups of flour.  Since our cake was ingredient free due to multiple food allergies, we used even parts of two gluten free flours.

"Okay so we need one and one half cups of flour.  How many halves in a whole?"

"Two!"

"Okay, so here is the 1/2 cup measuring cup.  One scoop of the buckwheat, one of the rice... Now, we need 1/2 cup more flour, using both our flours.  What's half of 1/2?"

"One-fourth!"

"How'd you know that?"

"Oh, I just knew..."
..............
Later in the morning, at the library.

"Hey look Mama!  The [really big, red tiled] square is made up of smaller squares!"

"Very cool.  How many small squares make up the big square?"

"Three, six...nine!"

"So, you wanna hear something interesting?  When you hear someone call a number 'square' that's what it means.  Look, nine squares literally make a 'square' number!  Isn't that cool?"

"Mmmm hmmm..."
.............
After lunch.

"Hey, it's time for cake!"

"YAY!!"

"But we can only eat half.  How would you cut it in half?"

[Girl gestures a line from the middle of one side to the middle of the opposite/parallel side.]

"Well, that's one way.  How else could you do it?"

[Girl pauses and then gestures diagonally from corner to corner.]

"Excellent!  But I have one more idea.  How many squares made up the big square at the library today?"

"Nine."

"So, I"m going to divide the cake into nine parts.  Here's one for you, and one for me. We'll have more later, after dinner."
...........
After dinner, and after one more piece each for the child and myself.  (Papa doesn't like chocolate, so no cake for him!)

"Mama, can I have a little more cake?"

"Hmmm, maybe.  We're only going to eat half of the cake -- let's see if there's any left....  How many pieces did we have to start?"

"Nine."

"What's half of nine?"

"Four and a half."

"So, together you and I have eaten four pieces.  There's half a piece left and you can have it!"

Ladies and gentlemen, I present to you: half a cake!

Chicken or the Egg? A Math Integration Tale

I recently had the opportunity to have my teaching work critiqued by a group of colleagues.  They viewed a ten-minute video I produced which illustrated what success looks like in my classroom. The feedback I received was all at once enthusiastic, thought provoking and puzzling.   

I teach elementary students the elements of percussive dance and then, within a structured framework, give them the freedom to create their own percussive patterns.  Along the way we use and talk about a lot of math which both describes their patterns and informs their creative choices.  It seems straightforward to me, so I think that is why I was flummoxed by a question they all had:

“When your students are choreographing their percussive dance patterns, how much of that activity is about their math understanding?”

I can answer that question.  The answer is, “All of it.”  I don’t see a separation between the two.   In fact, I think the dance and the math are essentially the same activity.

Here is an example:  a video of some traditional Irish figure dancing with accompanying percussive footwork.  You only have to watch a minute of the dancing to notice it is full of geometry and symmetry and all sorts of other wonderful kinds of math:

The shifting, curving patterns move through space and time while undergoing symmetrical transformations.  The dance choreography explores permutations and combinations of moves and steps by arranging and rearranging dancers at a dizzying rate in time to the music.  The footwork traces invisible maps on the floor.  The math in the percussive footwork is a reiteration of the figure dancing, but on a smaller scale with more specific and precise patterns.  Unsurprisingly, precision is a hallmark of mathematics which has, by a popular meme, been called ‘the science of patterns’.  

All this is well and good, but what my colleagues really wanted were more specifics about my evaluative criteria.  How exactly do I gauge my students --within the medium of percussive dance or with regard to the math?  Again my, possibly controversial, answer:  Both.  

And a question back: Why do we think of them as separate activities?  I think part of the issue might have a lot to do with how we, on the whole, perceive mathematical activity.

Generally conceived, dance is a three-dimensional, kinesthetic endeavor.  Math is rote memorization of algorithms and concepts and inhabits a two-dimensional symbolic realm.  Everything we’ve learned in school bears this out, except that it’s really not true!  When I started to really investigate what it means to do math I found that it’s completely different than what I did in school when I was a kid (and you too, probably).   

And, as I dug deeper, I also realized I had been thinking mathematically all my life – I just never recognized it as a mathematical activity.

One of the things I’ve come to realize is that, really, people who do math don’t spend a lot of time plugging numbers into memorized algorithms.  Instead,they formulate and/or approach questions that don’t have immediate solutions.  They spend time thinking, talk to others, sketch out ideas on napkins (or whatever), and build models.  And then, when they think they’ve got something that resembles a solution, that’s when they start writing it down.  The notation is the end result of a process of questions, trial and error, and conversations.  Sounds a lot like what we do in Math in Your Feet, actually.  Take a look:

When I first started wondering about whether or not there was math in the dancing I did with students, I knew I needed an interpreter, someone who really understood math and how it was taught to children. I was lucky to be connected with Jane Cooney, a classroom teacher with deep experience and love for teaching math.  Our collaboration in creating Math in Your Feet included long discussions about the best ways to retain the integrity of both content areas.  We weren’t going to make up the dance to fit the math and I wasn’t going to make up the math to fit the dance.

We didn’t and I haven’t.  There was no need. There is enough overlap between the two that, if you hit it right, you often can’t tell where one starts and the other ends.  However, I have consciously created specific lessons to identify and learn the math that we’re going to use in our dancing.  Not only is math a tool we need to understand in order to use it properly, but I think it’s also important to know exactly how math is involved in our physical and creative work.     

Like the old chicken/egg conundrum, it really doesn’t matter which one comes first because they’re both part of the same process.   And that is why, when I watch my students share their work throughout the week, I can see clearly if they have both the dance and the math and to what degree.  But that’s another story!

[This story originally posted in the Teaching Artist Journal's ALT/space, 11/28/12]

Morning Math

I'm really enjoying math in the morning.

There's something fresh and new and hopeful about mornings lately and, even though I'm not doing anything ground breaking, I'm really enjoying how connected everything seems to be these days, mathematically speaking. 

In mid-October I mapped out a basic math plan when it was clear the girl (now 7.5) needed and wanted more and different kinds of math challenges.  I decided to call the plan 'algebra' because, from what I've read, algebra combines a number of skills and concepts that we have to start learning anyhow at the primary level.  And, because the girl has often balked when I introduce new math stuff, calling it algebra motivates her to give it a try; 'algebra' is a big kid skill, and she really, really wants to be big.  

So, in the morning it's been fun to open my math folders and give my lovely child her choice of math activities.

Solve for x (conceptualizing equality and sameness, sums and differences) or math card games (3 digit mental sums and differences)?

Christmas themed beginner Sudoku puzzles or Factor Dominoes?

Growing patterns or Bean Soup (fractions, multiplication, division)?



















 "What's division, Mama?"

"You know, like when you wanted to see if you were halfway through your reader.  There were 96 pages and you figured out in your head that half of 90 was 45 and then you..."

"Oh, yeah."

Later that morning at the science museum she built this:

"Hey Mama, look!  I got the water to cover the whole area!"

"Cool!  That's an example of division, too.  The water is being distributed evenly across the table."

So, here I am, trying to connect our morning math time to the bigger math picture that I'm constructing in my brain.  Like algebra, as I've mentioned.  What is algebra???  I did some algebra when I was in high school, but failed literally and horribly (although I aced geometry).  But why let that stop me?  After some research I decided that the concepts of balance and sameness, solving for unknown quantities, and growing patterns were all pretty darn interesting and relevant whatever we called it and away we went.

We began by building and analyzing growing patterns from some pattern starters I found.  Here's one she came up with on her own:

And, when we do the occasional worksheet (the one below is skip counting/multiplication) I look for ways to extend the activity forward, even if I'm not completely sure I'm right.  The sheet below was for figuring out numbers of feathers on different numbers of hats, legs on rabbits, and petals on flowers.  She got the skip counting patterns easily, so I described the data another way by saying:

"The number of hats you have is multiplied by three feathers per hat which gives you the total number of feathers," while writing h x 3 = f.  The next two examples she talked herself through the whole thing and wrote it down.   But maybe it would have been better to say, "The total number of feathers you have is equal to the total number of hats multiplied by 3"??  Writing all this out has me thinking of another way I could have done that, but I think sometimes its okay for me to muddle through stuff like this.

And, if skip counting is coming easily, why not challenge her to fill in the chart backwards instead? Ooooh, that was cause for consternation, but she had been saying "This is easy, this is easy, " all morning.  I know for sure she needs more experience with subtraction, I know being able to invert a procedure or concept is an important skill, and I also know that if something is too challenging she'll just give up.  So, here was an opportunity for deepening her experience with a skip counting chart and providing just the right amount of challenge at the same time. 

Filling in the charts backwards really made her think.  That's what she wanted, right?

Anyhow, it's nice to be entering a stage with her where we can sit down and 'do math' outright instead of trying to leave it around the house for her to discover, although I'm keeping that strategy in my kip for now!  More than anything, it's been lovely doing math, in the morning, with the winter sunlight streaming in, with a child who is finally (finally!) allowing herself to be interested and excited about exploring this mysterious and wonderful subject.

Shhhh...Sneak Peek!

My new website is up in its temporary home waiting to have the Math in Your Feet address pointed to it.  In the mean time, I thought you might like taking a look.  It's got a ton of new features including an interactive News page and the brand new Math by Design section.  But the best thing of all is...

...a new two-minute Math in Your Feet video!!

I'm about to swoon, truly.  I've had the footage for a year or more, but only recently found the golden key that unlocked its riches. 

While you're there and if you're so inclined, I'd love to hear any feedback or comments you might have.  You can leave your thoughts here or in the comments of the News page.

You can find the site (for a few days) HERE.  

New Math Game: Factor Dominoes!

Lately I've been looking for different ways for my seven year old and I to conceptualize multiplication. As has happened many times before on our math journey, this graphic showed up at just the right time (albeit somewhat circuitously through the excellent influence of the Math Munch blog).

 

 

My favorite thing about it is that it's not about numerals; when I look at factoring trees I can make some surface sense of them, but my mind goes numb pretty quickly. In this visualization, however, there is an incredible connection to shapes and grouping. I find this visual especially well-suited for kids in general and at least this adult specifically.

Last night I printed out the graphic and left it advantageously on the kitchen counter. I thought maybe my kid might be interested but was truly surprised by her reaction when she found it this morning.  It is probably the first piece of math my daughter has ever admitted she was excited to know more about, which is saying a lot.

She wondered what it was about so we looked it over together.  At first it was basically 'count the dots' and notice that each configuration was one more dot than the one before. Then, in the same way we tackled the 100's chart last winter, we started looking around and noticing things: The ring of seven dots on the far right column has multiples of seven underneath it.  The 6 shape shows up two more times on a descending diagonal. It's fun just to look and talk about what you see.

It's the geometry of the design that really shows the relationships between numbers. And, even though this was not meant to be a multiplication chart, it's probably the best one I've ever seen.

All our talking and looking got my mind spinning. What if...what if I made little playing cards out of each factorized number? What kind of game would it be? 

I was about halfway through constructing the cards when my big AHA! moment hit. As I made and sorted them one by one it became completely clear to me that the integers 1 through 7 formed shapes that were echoed in the other factorizations.  As an attempt to organize my growing pile of cards I laid out a top row of 1 through 7.  But where to put the other cards? For example, 5 is a pentagon made out of single dots and 10 is a pentagon group of two dot groupings. Where does it belong?  The 2's column or the 5's column? This kind of question is at the heart of the new game.

Here's how my daughter decided to sort them in a 'get acquainted' activity before we started playing:

As we went along I refined the language she needed to help her make her choices. Was she going to place a particular card based on its large grouping (outer shape) or the smaller groups? As you can see above, there's a 5 shape of 3s in the 3 column, because the smaller group is a match to that number. But, every other 5 shape is in the 5's column. She's also got a 7 shape in the 3's column for the same reason -- the smaller grouping matched and, ultimately, the whole 3's column is consistent on that criterion.

For some comparison, here is how I sorted the cards, earlier in the day. I was trying to match to the category of 'outer shape':

I'm not sure I got it the way I wanted it, but no worries.  There is probably no one right way to sort these cards and the activity in itself makes for some really interesting thinking and conversation.

After she familiarized herself with the cards we started in on the new game which I'm calling Factor Dominoes (with a side of Scrabble). The title alone should give you clues as to the game's aesthetic and procedure, but here's how to play:

Split the deck equally between two players. Player 1 puts down the opening card. Player 2 tries to find a match. If Player 2 has no match the card is put aside face up for future use and play returns to Player 1. You can find a match either by outer grouping/shape (triangle, square, pentagon, weird six shape and seven ring) or by similarity between the small dot groupings. In our game we also matched 'echoes' -- small groupings that are the same shape as another number's outer shape.

For example, in the picture below the first card is a 5 shape with small groups of 2.  The 6 shape next to it works because even though it's a different shape it also is comprised of 2s. And, the card directly below the first card also works because the smaller groupings of 3 match the 5 shape of the larger grouping. Make sense? 

Here's another example: The top line of matches have the 3 shape in common. The bottom row connects to the top with small groupings of 4.

And, here's a picture of a couple more interesting matches.  See if you can figure out our reasoning on this section of the game:

Play the game until there are no more cards. This is a cooperative/conversational game but feel free to give it a point structure if you like. You can also make the game bigger and more complex for older students -- just cut out more factors and make more cards! That's what I'm going to do for our next round of play.

Here is our completed first game:

 

Based the exponential growth of my personal understanding of primes and factors, gained in just one short day, I am firmly convinced that a wide range of ages, experiences and abilities can get something of value out of this game. 

My seven year old was perfectly challenged as we focused on groupings, but what if you added the prime numbers beyond 7 into the mix? How would that deepen or change things? What about adding exponents as a match category? What if you figured the value of each card and matched them in sequences (like {25, 26, 27, 28...} or {4, 8, 12, 16...} or even a sequence of primes, in order)?

If you do play this game PLEASE let me know how it went and what other ideas you have for it.  And, please do consider joining us on the Math in Your Feet Facebook page. We're having a good time over there!

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Malke Rosenfeld delights in creating rich environments in which children and their adults can explore, make, play, and talk math based on their own questions and inclinations. Her upcoming book, Math on the Move: Engaging Students in Whole Body Learning, will be published by Heinemann in Fall 2016.