Did you have early math interests or facinations that grew with you into adulthood? What about your kids?
In his seminal book Mindstorms, Seymour Papert recounts the story of how a set of gears captured his imagination, first as a two-year-old and, later, as a school boy when he used the image of and experiences with the gears as a model for visualizing multiplication and fdifferential equations. James Tanton talks about a tiled ceiling in his childhood bedroom inspiring him to ask "What if…?" questions, which he later found out involved the mathematical idea of parity. As a child I was completely enamored with geometric design coloring books; as an adult I was drawn to and mastered a highly pattern-based dance form and found "the pattern that connects" (Bateson) between my art form and elementary mathematics learning.
Question: If you had a childhood fascination with a 'beautiful math object' that grew with you into adulthood (even if you didn’t know it was mathematical at the time), what was it that fascinated you when when you were young and how has it continued to influence your life into adulthood? In a related vein, how have your children's early math interests blossomed and/or expanded over the years? Is there one object, activity or idea that keeps showing up in different ways?
I found a book at the library about (simple) topology, donuts and coffecups! Then later Martin Gardner's wonderful math books.
My family had a set of unit blocks. I don't know their origin. They were painted, with a smooth finish. The square ones were blue, the one twice that size (a rectangle the size of two side-by-side flat squares) red. The corners were rounded off down the length of the block. I think half a red one (the long way) was orange, and half a blue one was green....maybe :).
I remember, vividly--not vivid in the sense of being able to picture it visually, but a strong recollection of the thought process that was going on--lining them up on top of each other, showing/exploring/verifying (what do you say there?) the relationships.
This would have been when I was three or four. One of my few memories from that far back.
I love hearing about early math memories! Here a pretty early one of mine: I must have been no older than 2 or 2 1/2 b/c I was sleeping in a crib. I remember waking up one night, everything quiet and calm, the wind rustling the curtains through the open window. And I remember *very* clearly that there were figures of a marching band (a drummer I remember most) marching across the curtain. I have a visceral memory of recognizing a pattern as it repeated across and up and down the curtains. :-)
My Mom had a collection of buttons, 4-5 handfuls, that I loved to play with. I would sort them by size, color or number of holes, what they reminded me of, what animals they'd belong to if animals needed buttons... I also pretend-played button beauty contest and button battles. In a beauty contest I'd line up buttons in groups of 3, 5 or 10 (since Mom always added/removed buttons from the box, I never knew ahead whether the total number of buttons was divisible by these numbers and I think it helped me with multiplication later on). I'd choose one beauty queen from each group to advance to the next round and repeat the process. Button battles were always one-on-one (so it helped with odd/even) or kind-of-the-hill style (where every button had to go against the last battle's champion and, if a winner, would become the champion and go against the remaining buttons). So lots of rules :) I remember I could be busy with these for hours.
Later, around 4th grade, I was shown this game where you draw a 10x10 square on a grid paper. Then you start filling it in with numbers from 1 to 100. You can start anywhere on that square. But you can move only like the horse moves in chess (3 up 1 over or 3 over 1 up). It gets very difficult once you get to 91, 92, etc and getting a perfect 100 was particularly tough.
I am having my teenaged son join me with this course, and as soon as he read your description of the 10 x 10 game here, he immediately made a grid on a piece of paper and went to work!
Here's another example of how useful beautiful objects can be in math learning. This is an excerpt from Fawn Nguyen's latest post which can be found at: http://fawnnguyen.com/2014/02/10/20140210.aspx
"Instead of waiting for When Will I Ever Need This? to come up in the middle of our favorite lesson, let's address it on Day 1 of school. Let's show (not tell) our students how beautiful and useful math is. How fun and challenging learning math can be and should be. Show them a math video and read them a passage from a math book that make us cry."
Puddles, I was absolutely taken with the puddles. The surprising thing was that an object's reflection moved when I moved while it was not mine but the object's.
I went into math because it was the only thing I was good at in high school. I struggled with calculus and everything after, muscling my way through with gritty determination and luck, until I finally gave up. I never really understood it. Eight years later I finished college with a liberal arts degree. Seven years after that I took a job teaching middle school math and was required to use the Connected Math textbook. A very student-centered problem based approach to mathematics.
I learned more about math teaching that class than ever before, for the first time in my life math was not about solving problems, but making sense of the world.
Interesting question! I had early math interests and fascinations without realizing they were math. I remember lolling about, looking up from an upside-down position wondering what it would be like to traverse the ceiling. I had an aunt and uncle that encouraged me to look very closely into flowers.
My mom had a Czech friend, who brought her beautiful crystal and glass necklaces. They became toddler toys, and soon a box of beautiful loose crystal and glass beads. I spent hours and hours arranging them into necklaces and patterns of all kinds. By now I know I was playing with sequences, symmetry, and quantity... As a grown-up, Hesse's book The Glass Bead Game is one of my favorite stories about education. The metaphor of glass beads is hugely meaningful for me to this day.
Just heads up that the book isn't really about manipulatives of any sort. It's about abstract philosophy. And human stories.
What strikes me about this idea of evocative objects is that it's often an internal, private experience for the child. It's also slow. In the Preface to Mindstorms, Papert says:
"If any 'scientific' educational psychologist had tried to 'measure' the effects of this encounter [with the gears], he would probably have failed. [The gears] had profound consequences but, I conjecture, only very many years later. A pre- and post- test at age two would have missed them."
If this isn't an endorsement for math learning in a rich context, including classrooms and homes filled with mathematical objects and things to wonder over, I don't know what is.
"Evocative objects and experiences" are often so internal and private that they never surface. It's only when there is an environment that is prepared for "wonderment" and when there are adults around to observe and add to that environment accordingly that children know that it is OK to be curious, ask questions, and marvel at what they don't know or understand (I'm Montessori influenced). An all too common childhood experience is when a child feels bothersome and always underfoot. He/she begins to dismiss his/her own moments of intuition, insight, "magical" thinking as aberrations.
YEAH! to "classrooms and homes filled with mathematical objects and things to wonder over."
