Proof before high school - remember to ask "Why?"
You might think, "Proof? You need it before high school geometry?"
Sure! But, we are talking about a different form of 'proving' here.
What IS proof, first of all? It is something that the person hearing or reading the 'proof' will become CONVINCED that whatever you're proving is, indeed, true.
You will want to convince your youngsters or students that what YOU ARE telling them, is indeed true. That is 'proving', in a general sense. But it doesn't have to be on the same level as later.
For example, when you are showing them how the multi-digit multiplication works and WHY it works (2 × 371 is the same as 2 × 300 + 2 × 70 + 2 × 1), you are proving - or maybe we should say "justifying".
When you take fraction manipulatives and demonstrate why 5 × 2/3 is 3 1/3, you are 'proving' - or demonstrating. (You take 5 times 2/3. You combine the thirds until you get 3 wholes and one third.)
Oftentimes diagrams or picturial illustrations work as 'proofs' in grade school. When students "see it", they become convinced. For example, using algebra tiles to show factorings in pre-algebra.
Those proofs are not as "rigorous" proofs that mathematicians do, but they are important, nonetheless. That way students get used to demonstrations as to WHY something works, and DON'T get used to "announced math" where rules are just given as step-by-step instructions without much explanations.
They can keep that small voice inside alive that says, "Why is that?"
Then, after you as the teacher have done it, change turns. It's time for YOU to ask WHY.
A good place for this is after the student has done some problems: ask why he/she did it this way.
"You added right. Now tell me WHY did you put that 2 up there above the other numbers?"
And, of course you will take special notice of the problems asking the student to justify his/her reasoning.
It doesn't have to be anymore complicated than that. Just don't forget it amidst all the drill and "you need to complete the worksheet" stuff. Take time to reflect, sometimes, and consider the "why".
See also the previous posts: What is proof? and Two-column proof versus paragraph proof.
Tags: philosophy, math
Sure! But, we are talking about a different form of 'proving' here.
What IS proof, first of all? It is something that the person hearing or reading the 'proof' will become CONVINCED that whatever you're proving is, indeed, true.
You will want to convince your youngsters or students that what YOU ARE telling them, is indeed true. That is 'proving', in a general sense. But it doesn't have to be on the same level as later.
For example, when you are showing them how the multi-digit multiplication works and WHY it works (2 × 371 is the same as 2 × 300 + 2 × 70 + 2 × 1), you are proving - or maybe we should say "justifying".
When you take fraction manipulatives and demonstrate why 5 × 2/3 is 3 1/3, you are 'proving' - or demonstrating. (You take 5 times 2/3. You combine the thirds until you get 3 wholes and one third.)
Oftentimes diagrams or picturial illustrations work as 'proofs' in grade school. When students "see it", they become convinced. For example, using algebra tiles to show factorings in pre-algebra.
Those proofs are not as "rigorous" proofs that mathematicians do, but they are important, nonetheless. That way students get used to demonstrations as to WHY something works, and DON'T get used to "announced math" where rules are just given as step-by-step instructions without much explanations.
They can keep that small voice inside alive that says, "Why is that?"
Then, after you as the teacher have done it, change turns. It's time for YOU to ask WHY.
A good place for this is after the student has done some problems: ask why he/she did it this way.
"You added right. Now tell me WHY did you put that 2 up there above the other numbers?"
And, of course you will take special notice of the problems asking the student to justify his/her reasoning.
It doesn't have to be anymore complicated than that. Just don't forget it amidst all the drill and "you need to complete the worksheet" stuff. Take time to reflect, sometimes, and consider the "why".
See also the previous posts: What is proof? and Two-column proof versus paragraph proof.
Tags: philosophy, math
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