As I mentioned in a previous entry, I have just recently begun homeschooling my daughter Kylie. Naturally, I have a very strong interest in her mathematical development. I want her to feel that all career options are available to her and the truth is that the lack of mathematical understanding is all too often the cause of career "doors" slamming shut. I want her to feel as if, should she choose, she could be a mathematician, a chemist, a physicist, an economist, an architect, a computer scientist, a pharmacist...you get the idea. Math plays an important role in many, many jobs. Those of you who have had me as an instructor know that I believe that if you are not running into situations that make you stop and really think about what you know and how you might use that information, then you aren't really learning in a way that is fostering intellectual development. So, we began the school year with what I would describe as "challenging, yet achievable" problems. Much to my surprise, Kylie was not very happy. Kylie has always loved math and enjoyed tackling difficult problems, so I had not expected this response. Part of the difficulty arose from the transition from school to homeschool. She was used to strategy shares when she could spend more time pondering other people's strategies than in sharing her own thinking. She is a bit of an introvert, so initially she found the idea of the "spotlight is just on me" to be overwhelming. Most of these issues have worked themselves out, but I have spent some time lately thinking about how to keep her love of math alive through the "tough spots."
I have just finished reading Jo Boaler's What's Math Got to Do With It? ( Subtitle: Helping Children Learn to Love Their Least Favorite Subject--And Why It's Important for America). In this book she mentioned that many famous mathematicians did not develop their love for the subject in school, but rather from someone who shared mathematical puzzles with them. The mathematical puzzles were intriguing and pulled them in to mathematics in joyful ways. So, I introduced Kylie first to the container problem (see sample puzzles at the end of this entry). She was riveted, frustrated, and ultimately thrillingly victorious! I do realize that math instruction cannot consist of just a steady diet of puzzles. However, I have come to see the need for them to play a role in the big picture of mathematical instruction. I don't pretend to have the "perfect" plan for approaching math, and I'm sure that this will evolve, but currently we have a 3-phase math time. Initially, we have a series of mathematical exercises that have been specially chosen to highlight certain mathematical relationships, suggest efficient, sense-based strategies, and/or see the use of particular mathematical models. If you are a former student, you may recognize this as the "computational minilesson" that Fosnot/Dolk talked about in Young Mathematicians at Work (by the way, if you liked those books, did you know that they now have supplemental curriculum available? Check it out at http://www.heinemann.com/ ). Following this time of strategy development, we do the concept-development phase of instruction. For this portion, I am using the Japanese curriculum that I mentioned in an earlier entry as a general guide. I do supplement it depending on Kylie's needs. The final portion of the class is usually choice time. I have given her a selection of mathematical puzzles and extended problems to explore. She chooses which ones she wants to do, and approaches them in any order, depending on how they intrigue her. Adding this component has had a hugely positive impact on her attitude. I also give her a "menu" of open-ended problems related to our current topic part way into the unit. She has to have these done by the end of the unit, but this still allows plenty of time for her to explore other problems.
These three phases allow Kylie to: 1) develop her strategies and grow in her understanding of number relationships and mental math; 2)move her forward into newer concepts in a way that builds on her previous schema, and; 3) allows her to see herself as a true mathematician tackling intriguing problems.
Container Puzzle:
Given a five-liter container, a three-liter container, and an unlimited supply of water, how do you measure out four liters exactly?
The Rabbit Puzzle: (also taken from Jo Boaler's book)
A rabbit falls into a dry well thirty meters deep. Since being at the bottom of the well was not her original plan, she decides to climb out. When she attempts to do so, she finds that after going up three meters (and this is the sad part) she slips back two. Frustrated, she stops where she is for the day and resumes her efforts the following morning--with the same results. How many days does it take her to get out of the well?
The Chessboard Problem:
How many squares are on a standard 8 x 8 chessboard? (Keep in mind that this is referring to squares of all sizes). How can you know for sure if you have found all of the possibilities? Can you generalize a method for finding out the number of squares of any chessboard of with different edge lengths? (For Kylie, I printed off sheets of empty chessboard clipart that she can use to write on, if she wishes. This is the problem she is currently working on. I haven't seen her work. She prefers to do her thinking on these problems on her own as much as possible.)
For some online mathematical puzzles, check out these links. The first one is interactive:
http://www.math.com/students/puzzles/puzzleapps.html
http://www.jimloy.com/puzz/puzz.htm
http://thinks.com/puzzles/loyd/loyd.htm
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