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Today is all about “Food for Thought”.
Let’s do an experiment.
Pretend you are at the grocery store.
You have $9.50 in your wallet and you forgot your credit card. You want to buy a watermelon which costs $4.25 and a gallon of milk which costs $2.99. Will you have enough money?
There are lots of ways you might solve that problem in your head.
Maybe you looked at those prices and thought “2.99 is almost $3. So… $4 + $3 = $7. Then add in the quarter and you get $7.25. But take away the penny from the $2.00/$3.00 conversion, and my final total is $7.24. So yes, I have enough money.”
But can you still go to Starbucks afterwards? How much money will you have left over?
Now you might be thinking “$9.50 – $7.24 = about $2.25. I can afford a cup of coffee, but not a mocha.”
What you just did was real life math.
I bet your accuracy was fantastic.
But now let’s do another experiment.
Pretend you are in middle school. You are crammed into one of those super uncomfortable chairs with the attached desks. The teacher is some old guy up front wearing a weird tie. There is a vague smell of tuna-melt wafting through the room from the cafeteria next door.
You are taking a really important test. NO MISTAKES ALLOWED! Here’s the problem:
950 – (425+299) = ?
Now what are you going to do?
If you are like most people my age or older, you might have reached for a paper and pencil. Probably you stacked those numbers up and started borrowing and carrying. It might even be possible that your accuracy went down.
It’s too bad you couldn’t unleash your inner math instinct.
One of my favorite books about math is called The Math Instinct: Why You’re a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs) by Keith Devlin. In the final chapters of the book, Dr. Devlin shares research about this phenomenon of grocery store math vs. school math.
In real life people form their own strategies for solving problems and tend to be pretty accurate in their calculations. In school though, those same people can be hit-and-miss as they attempt to crank out traditional algorithms.
Maybe you are better at math than you thought.
Give yourself permission to let go of the notion that there is one right way to solve a problem.
Give your children permission to explore and discover strategies that make sense to them.
Then imagine the possibilities…
If you can already do grocery store math in your head with kick-awesome accuracy, what do you think might have happened to your own math ability if you had been encouraged to follow that type of problem solving as a young child? Would you be doing algebra in your head? Would calculus be fun?
When a teacher forces a child to do something over and over again that doesn’t make any sense to them, how is that helping?
In my ideal world, “Drill and Kill” would be illegal.
The real art in teaching math to children comes from asking them questions about their own thinking and then listening. “How would you do this? What makes sense to you?”
It’s really hard to do that in the classroom, especially if you are responsible for 30 kids.
But the next time you are at the grocery store with your own son or daughter, try asking them if you have enough money to buy ice cream. Then listen to what they have to say…
Math Boot Camp for Moms, Day 2
Reading Leonardo and Steve: The Young Genius Who Beat Apple to Market by 800 Years, by Dr. Keith Devlin is easy, entertaining, and educational. Those are three qualities that make Leonardo and Steve a good choice for a leisure-minded non-fiction enthusiast like me. You don’t have to be a math genius to understand this book either. In my case, that’s a good thing!
The crux of this book is the parallel between Steve Jobs and Leonardo Pisano; how they seized upon, changed, and communicated the inventions of other’s and sparked financial and personal computing revolutions. Leonardo (also called Fibonacci) did this in 1202 by writing the book Liber Abbaci which introduced Hindu-Arabic numbers to the businessmen of Pisa, and explained how these numbers made accounting much easier than the Roman numerals they were using.
All of that would be interesting in its own right, but to me as a former elementary school teacher and participant in the world of gifted education, there are some other random things about Leonardo and Steve’s stories that really strike me.
Dr. Devlin briefly mentions that after the publication of Liber Abbaci, arithmetic schools sprung up over Italy, where maesti d’abbaco would teach students the new Hindu-Arabic methods. He says that these schools “followed a specified syllabus, typically comprised of reading and writing in the vernacular, arithmetic, geometry, bookkeeping, and occasionally navigation.” (Loc 274, 29%) A specified syllabus? That almost sounds like Common Core Standards from the Middle Ages!
The other section of Leonardo and Steve that I found fascinating was this description of what it is like when a computer programmer gets lost in thought while at the computer: “Once you get into the project, it develops a life of its own. You find yourself in what is often referred to as “The flow”. Time stands still, and the mind is able to cope with any amount of fine detail. Indeed, it does not seem like fine detail; at that moment that design or that piece of code is all that matters in the world.” (Loc 235, 25%) I thought that was good explanation of what happens to gifted people in general when they get super-focused on a project. In fact, perhaps it is the gifted brain’s ability to focus on something for a long time (the so-called 10,000 hours effect) that leads to achievement.
A final thought about Leonardo and Steve is that it could very well be read as a sequel to Outliers: The Story of Success by Malcolm Gladwell. What would have happened if Leonardo had not been born into the nobility? What would have happened if his father had not taken him to Algeria where he encountered Hindu-Arabic numbers? What would have happened if Steve Jobs had not lived in Cupertino? You could very well make the argument that these men would not have made such a big impacts into modern lives if they had not been born in the right places and the right times.
P.S. I have no idea if I annotated these page numbers correctly or not, so I apologize if I made citing errors. This is only the third eBook I’ve read and it took me a good deal of time figuring out how to use the highlighting function. If I don’t make a concerted effort to keep up with technology I’ll end up someday as the grandmother who doesn’t know how to turn on her TV! 🙂
This weekend Bruce(6.5) and I had fun building real life triangles. I was inspired to try this activity after reading Keith Devlin’s thought provoking book, The Language of Mathematics. Unlike The Math Instinct, The Language of Mathematics focuses more on the history of mathematics rather than best practices in teaching mathematics.
Dr. Devlin wrote The Language of Mathematics for the general reader, but I found it quite challenging even though I was an A- student in mathematics all the way through college level Calculus. I wish I had read this book back in high school, because it might have helped me knock off those minuses! On page 44, he discusses 3-4-5 triangles and ancient builders.
A 3-4-5 triangle is also mentioned in a children’s book Bruce and I read recently by Avi called The Barn. That too, was an excellent book although I would not recommend it for children under ten (oops) because it dealt with a really depressing storyline. In The Barn, the children use 3-4-5 triangles to create the right angle of the barn they are building for their father who is dying after suffering from a stroke. Here is how we built our own 3-4-5 triangle:
First, Bruce measured out 12 segments on masking tape that was folded on itself so it wasn’t sticky. He used Legos as his nonstandard unit of measurement. It would have been better to tie knots in rope or yarn at 12 equal lengths, but I knew that tying knots with a six year old would have added about ten minutes of utter frustration to the activity, and fine motor skill improvement was not my learning objective. Plus, I couldn’t find any rope. 🙂
At the 3 and 7 marks, we pulled the triangle tight. This picture doesn’t do the triangle justice, because I was also trying to hold the camera, but we did end up forming a right triangle with sides that were equal to 3-4-5 Legos. This triangle was also scalene because all three sides were different lengths.
After building our right triangle, we built a giant Isosolese triangle on the ground for fun. We also got out our protractor and measured angles, but as you can probably tell from the picture, since we had built our triangle with masking tape instead of rope, the angles were off by a few degrees. Bruce had previously had trouble remembering what an Isosclese triangle was, so hopefully this helps him remember. You can fit an “I” in an Isosceles triangle. Hopefully he also remembers what a right triangle is too.
In case you are wondering, I’d offer a hesitant yes to this question regarding Jenna(29 months). She correctly identifies quantities of three about 80% of the time. There is no way she is ready to move on to four yet, despite her ability to “count”, i.e. rattle off numbers without correspondence. I got a pretty accurate understanding of where my daughter’s thinking currently resides, by trying out lesson 1 from Right Start Level A. You can download the first few lesson plans yourself for free and give it a try with your own two year old.
I wish I had known to try this experiment with Bruce (6.5) when he was two, because of course I’m now uber-curious what his thinking was back then. When did the quantities three and four really solidify for Bruce? When will they solidify for Jenna? The Psychology major in me is going to be periodically checking to figure this out. Looking forward in lessons 2 and 3 of Right Start Level A, the other two abilities to monitor are A-B-A patterns, and sorting.
The reason why I decided to investigate all of this to begin with, was that I have just finished reading Keith Devlin’s wonderful book The Math Instinct: Why You’re a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs). This book was well written, meticulously researched, and thought provoking. The final chapters reaffirmed everything I learned about Constructivist math in my professional development as a teacher. The research he presented about young infants understanding the quantity of two was especially fascinating. I had previously been proud of Jenna understanding the quantity two. Now I realize that’s no big deal!
I’m going to continue on reading more of Keith Devlin’s books. I’m not a “mathy” person myself, but he writes in a way that is easy to understand… even for someone who hasn’t studied Calculus since 12th grade. 🙂