“Art of Inquiry” circle, Rockville, MD : “Art of Inquiry” circle, Rockville, MD Julia Brodsky
“ Art of Inquiry”, Founder
www.artofinquiry.net
Julia Brodsky - Short bio : Julia Brodsky - Short bio Magnet math and science high school, Russia
St. Petersburg Polytechnic Institute, Russia
NASA – training astronauts in solving
off-nominal problems
Mom of 3 kids: 9, 7 and 3 years old.
Circle: Why and how? : Circle: Why and how? Why? Out of desperation
Inspired by the wonderful book by Alexander Zvonkin “ Math for little ones” ( in Russian, to be translated by MSRI ) How? Weekly 1 hr sessions (a warm-up, followed by a few harder problems)
One teacher, with a high school student-helper
Notes taken on students’ frustrations, approaches, mistakes, etc
Problems are presented in written format, with pictures if possible
Kids are provided with various manipulatives (checkers, blocks, clay, toothpicks, mirrors, etc), as needed
Student population : Student population Age: Group I (6-7 yrs), Group II (8-9 yrs)
Class size: 6-8 students per group
Cons:
Jumpy, noisy, no good manners ?
Limited math skills
Group I: addition-subtraction in the range of 0-20
Group II: learning multiplication table
Pros:
Curious and enthusiastic
Few mental blocks
Start early! : Start early! By upper middle/high school kids partially lose their innate ability to solve non-standard problems that they have in childhood, when every problem comes across as a non standard.
Play with the problem - without the pressure to get a specific result by specific time.
Make mistakes and enjoy them!
Learn to distinguish what is relevant, what is not.
Ask complex questions, solve multi-step problems
Adapt to new situations.
Work as a team; respect others’ opinions
Reflect on a solution
Problems kids love : Problems kids love It is amazing to observe that young kids tend to favor problems they do not know how to approach – aka “insight” problems. They readily spend time exploring, and share their solutions with friends and parents.
“Insight” problems also motivate kids to look for “insightful” ways of solving “regular” problems, approaching them from different angles, recognizing and avoiding clichés.
While I do not know how to systematically teach students to solve “insight” problems, my observations show that if kids expect an “ insightful” solution, they are much more likely to find one
“ Insight” class, Group I – Warm up : “ Insight” class, Group I – Warm up Continue the pattern Kids spent a lot of time analyzing the pattern and arguing with each other trying to persuade others in the rightness of their solution. The situation resolved when one of them put a pencil at the center of the figure and saw the “ reflection” feature. This was the “Aha!” moment I was waiting for. Symmetry concepts are very new for kids of this age. It is a good time to introduce them ( start with mirror symmetry, etc)
“Insight” class, Group I – Problem 1 : “Insight” class, Group I – Problem 1 It is known that two kittens weight less than their cat mom. Do three kittens weight more than their mom? My objective here is the analysis of the problem. Kids need to recognize the problem as undefined. A lot of discussion. Some said more, some said less, and they seemed to be very confident in their answers. Then Ben came up with his original idea – they are equal. I asked him whether the kittens were well fed or not. Both he and Ann said that the kittens were well fed. Here Marie and Art started to argue. This was the “ uncertainty” point that ultimately lead to recognition that they can not solve the problem the way it is given. We spent some time asking each other questions that are not well defined – silly activity with plenty of laughing.
“Insight” class, Group I – Problem 2 : “Insight” class, Group I – Problem 2 Move 2 matchsticks to get 5 equal squares. Kids got the matchsticks and sat down to work individually. It is amazing how quickly they found the solution. “This was an easy one” said Ann. “ Can we get more like this?”* *I expected this problem to be appealing to the kids - it is clearly defined, visual, hands-on and does not require special skills.
“ Insight” class, Group I – Bonus fun : “ Insight” class, Group I – Bonus fun Make a knot on the rope while holding both ends of the rope. It was a big hit. Everybody wanted to try. They would not leave the class after it was over. I left it as a homework problem.
“Insight” class, Group II – Warm up : “Insight” class, Group II – Warm up 8809 = 6 7111 = 0 2172 = 0 6666 = 4 1111 = 0 3213 = 0 7662 = 2 9312 = 1 0000 = 4 2222 = 0 3333 = 0 8193 = 3 8096 = 5 2581 = ? The younger you are, the easier it is for you to solve. One statistician used Excel to solve it, another PhD in Mathematics spent a while analyzing it... The kids saw the solution almost instantly.
“Insight” class, Group II – Problem 1 : “Insight” class, Group II – Problem 1 How to fry 3 pancakes in less than 8 minutes if you can only fry 2 pancakes at a time, and each side takes 2 minutes? In the course of our studies, kids grew suspicions of constrains. They tried “breaking the cage”. First responses were: “Turn the heat on!” , “Get a bigger pan!” , “Make thinner pancakes!”. I told them all of their responses were great, and introduced tighter constrains.
Many kids were flipping their palms and Eva tore 3 pieces of paper out of the notebook to experiment with them. In a minute, she ran out to the board, happy with her discovery. *While many teachers would see such responses as a nuisance, my observations suggest that kids greatly benefit from this type of constrain analysis.
“Insight” class, Group II – Problem 2 : “Insight” class, Group II – Problem 2 Tony has 3 bricks and a ruler. How can he measure the diagonal AB of the brick with that ruler?* This problem generated a lot of interest. Kids tried all types of approaches: “Please give us a saw!”; “ Let’s drill through the brick!”;“ Let’s make this brick out of modeling clay and then cut it!”. The last exclamation made them think of clay and snow impressions and of measuring the diagonal inside the impression. Even before I told them that clay and snow were unavailable, they were well on the way solving the problem with the help of 2 other bricks. * One dad ( an applied mathematician) happily solved it by using the Pythagorean theorem. His 8 year old daughter showed him a much easier solution ?
“Insight” class, Group II – Problem 3 : “Insight” class, Group II – Problem 3 Is it possible to put 5 checkers in such a way that each checker touches 2 others?
3 others? Kids easily found a 2D solution for the 1st problem (which I gave as a teaser). They started the second part of the problem in high spirits, but soon recognized that they needed to go 3D. Here they entered an elevated state of
“ almost there”. They meticulously checked each others’ solutions and pointed out the errors. Parents were eager to try themselves, too.
Summary : Summary It is never early to introduce the joy of thinking
It is enjoyable, educational and inspiring to both kids and teachers
Our society needs “out of the box thinkers” more than ever
Kids need to learn to search for and to recognize the feeling of the “insight” moments; to distinguish between knowledge and understanding.
Slide 16 : And now…
a problem for YOU
The Royal Prisoner : The Royal Prisoner Many years ago, a French king was taken as a prisoner by Spain. The Spanish king ordered to make the entrance to the throne hall so low that everyone would have to bow to get through. However, the royal prisoner wanted both to avoid humiliation and to show his despise of the Spanish king. What did he do?..
Our “insight” problems were collected from various sources, such as… : Our “insight” problems were collected from various sources, such as… B.Kordemsky, “ The Moscow puzzles”
R. Smullyan, ”The Riddle of Scheherazade And Other Amazing Puzzles”
A. Zvonkin, “Math for little ones (math circle diary)” (Rus)
www.puzzles.com
www.mathkangaroo.org