How can creativity be incorporated into elementary school mathematics?

Question

Creativity is the core of research mathematics. However, most introductory math consists of learning fixed rules to perform basic, essential mathematics.

Thus, for many elementary school students, math is perceived as a rote and routine subject. This doesn't turn everyone off; a lot of people love math into their teens. But creativity is not seen as part of mathematics.

How can creativity be introduced at the elementary level?

My teachers did this exceptionally well; one of my favorite memories involves the classic 'four fours' puzzle: using the symbol 4 four times and unlimited math operations, create all numbers from 1 to 100. I remember working on this with my grandmother.

What general principles can be used to promote creativity in elementary school mathematical students?

Answer 1

I have a distinct memory of my second grade teacher handing out a logic grid puzzle to the class one day near the end of the year. I remember thinking, "This is fun, but what does it have to do with math?"

Of course, logical puzzle-solving is an important kind of math, and should be explicitly included in the curriculum. Students of all ages should be taught about logic grid puzzles, tangrams, mazes, magic squares, sodoku, minesweeper, Rubik's cubes, towers of Hanoi and so forth in math class, as well as games such as nim, dots and boxes, sprouts, etc. Puzzle-solving is a basic mode of creative thought, and it needs to be explicitly included in the curriculum.

In addition to such explicit logic puzzles, teachers at the elementary school level should endeavor to include small mathematical puzzles that the students can potentially solve. Especially if they work in groups of two or three, fourth or fifth grade students ought to be able to solve a simple puzzle like

Find two numbers whose sum is 22 and whose product is 96.

We teach explicit methods for these things when they get to algebra class, but in elementary school students ought to be clever enough to figure out a puzzle like this without having any explicit method. Other puzzles like this include

How many different ways are there to make 36 cents in change? (I'm assuming U.S. currency.)

or

A solid, four-inch cube of wood is coated with blue paint on all six sides. Then the cube is cut into smaller one-inch cubes. These new one-inch cubes will have either three blue sides, two blue sides, one blue side, or no blue sides. How many of each will there be?

A general rule for such puzzles is that the children should not be solving them using some method or algorithm that they were explicitly taught. Math is not a rote subject, and puzzles like these are much more representative of real mathematics than multiplying three-digit numbers or reducing fractions to lowest terms.

Finally, in addition to puzzles, there are many possible geometry-based toys that children can use creatively, including polydrons, zomes, tangrams, and so forth. These could lead to all sorts of interesting classroom activities -- e.g. a contest to see which group can construct the most interesting closed shape (polyhedron) using polydrons.

Answer 2

Resources
You might like the book, Moebius Noodles, to use with young kids. I'd also recommend checking out math circles. This book is a delightful start. There's lots more. And you might share The Number Devil with the young people you work with. The book I'm working on, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, will also be helpful once it's out. (Actually, you're welcome to request a copy of the manuscript now. Just email me.)

One creative problem, that I'll be doing as a math circle with college students (starting next week), is the question of how the cards for Spot It! were created. I blogged about it here, here, and here. It doesn't require algebra, but is related to a field of higher math.

Principles
Your second question is interesting: What general principles can be used to promote creativity in elementary school mathematical students?

Perhaps this passage from my book illustrates what I see as the first principle, that math is more than arithmetic, and even young kids can see its other sides:
What is math? Most people think it’s adding, subtracting, multiplying, and dividing; knowing your times tables; knowing how to divide fractions; knowing how to follow the rules to find the answer. These bits are a tiny corner of the world of math. Math is seeing patterns, solving puzzles, using logic, finding ways to connect disparate ideas, and so much more. People who do math play with infinity, shapes, map coloring, tiling, and probability; they analyze how things change over time, or how one particular change will affect a whole system. Math is about concepts, connections, patterns. It can be a game, a language, an art form. Everything is connected, often in surprising and beautiful ways.

Second principle: If we want to promote creativity, we need to let kids know that math is playful. (I think those two principles probably sum it up for me.)

Answer 3

Have them measure their world in creative ways:

How much water does your bathtub hold at home? If the bath is nearly full, and you get in the water up to your neck, how much does the water level go up?

On a map, draw the route that you take to get to school. Draw the route that a bird would take flying directly. How long is the bird's route? How long is your route? Where are the routes furthest apart?

Draw a floor plan of your room at school, and a room at your house. Where are there circles, rectangles, squares, triangles or other shapes? What shapes would you like to have?

There's lots of opportunity for creative mathematical thought along these lines.