Food for Thought - Intuitive understanding of any mathematical idea is rooted in direct personal experience with the idea itself. Direct contact with the ideas behind the symbols brings meaning to the marks we make in the name of math and allows learners to deeply own the ideas behind those marks.

Full Contact Math
... a personal experience

8:30 am.  The buzz and chatter in the room subsides.
“Good morning…” says the announcer to the sixty-plus teachers sitting at tables throughout the room, each hopeful that the day will be worthwhile, bringing new light to his/her math teaching. 

Introductions are delivered and the guest speaker steps up to the microphone.  There is a brief pause.  “Has anyone here ever eaten a mango?” she asks.
Somewhat perplexed, the audience hesitates briefly before several hands go up.  Maria’s hand is not raised.  “So then, Maria, you’ve never eaten a mango?”
“No.” replies a young teacher at the back of the room.  The speaker addresses the rest of the group.  “All right… I would like the rest of you to explain mangoes to Maria so that she understands.  Can you do that?”
There is another pause.  Contributions begin slowly, but gain momentum and volume as the exercise continues, several people talking at once.

“… juicy”
“…more slimy than juicy”
“… sort of tastes like a peach”
“…but with an orange sort of flavour”
“… kind of stringy”
“…soft flesh”

“So, Maria, now do you understand about mangoes?” the speaker asks.
“Sort of… not really…” she replies.
The group is invited to try again.

“…sweet”
“… smells a bit like socks”
“… I agree that it’s more like a peach than an orange”
“… gets between your teeth”
“… sort of rich tasting”

Maria is obviously trying hard to understand but doesn’t look convinced.

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