Traditionally we give a concept or algorithm to the students and ask them to memorize, understand and use it. And by "give" I mean we serve it to them well done, fully baked, nothing left to do but eat it.
Piaget’s research (and subsequently others) suggest that allowing students to discover or create the methods is more effective than handing them the method and asking them to take it, eat it, no questions asked.
But how do you let them discover it?
Since the math we teach in middle school and high school is based on real numbers, every concept can be demonstrated with “plain” numbers. Which means it can be discovered by playing with numbers.
Calculators make this playing or experimentation fast – giving a student the ability to quickly see patterns and construct concepts.
Introduce the topic with numbers.
When you introduce a topic, give 10-20 “examples” of it with real numbers. Ask the students to play with them and notice any patterns they find.
Notice you’re creating an inquiry-based learning environment, an Inquiry Zone for Math Learning. Remember to maintain positive body language, ignore negativity, and super-validate what any student says.
Give the students the power!
Until the student decides differently, everyone is wrong. Even the teacher and textbook. They get to validate it for themselves – and they can do this with real numbers.
Ultimately, if they grow to be mathematicians, they’ll learn that verifying with lots of real numbers doesn’t mean “proving” it – but for the time being, this works fine.
Giving them this power lets them experiment as much as they need, and only as much as they need, to verify a concept for themselves.
Use the Play & Say method.
You’ve heard of the “Plug & Chug” method? You take a formula, plug in the numbers and chug through the arithmetic. Plug & Chug is a non-discovery based practice tool. The practice is good, mind you, but the formula is given, not discovered.
Remember, something discovered is more likely to be remembered than something given.
So use the “Play & Say” method. Each student plays with the numbers until he or she can say what the formula or concept is.
If you’re trying to teach a concept with this and one student discovers a different formula or concept. Run with it – as long as it’s mathematically sound. Don’t discourage the discovery of anything, even something not on the current curriculum.
If a student gets frustrated, don’t force them to discover it themselves. Give them the concept or formula and encourage them to experiment later with it.
If you find there is a big difference in how much time each student takes, send the experiments home with them. Give them five minutes at the beginning of class to play – the students who realize they need more time will have done more the night before.
How will you do it? Share your thoughts and experiences in the comments or via twitter:
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