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A funny thing happens when you celebrate student suggestions. More students want their suggestions celebrated!
In my calculus class, a student noticed that there was a pattern for the tangent slopes of the function \(y=x^2\) namely that the slope was always twice the input. He went on to hypothesize that for the function \(y=x^3\) the slopes would be the absolute value of three times the input. I named it after him (The Christian Hypothesis). [Note: I know the second part in incorrect, but we have a statement we can verify later -MrC]
Today, as we continued to explore slopes and derivatives (the function that finds the slopes of a function) the hypotheses came pouring out. The Mia/Daisy Joint Conjecture found that the derivative of a parabola was a line, the derivative of a cubic was a parabola, and the derivative of a fourth degree function ( \(y=x^4\) ) was a cubic function. The Bre Hypothesis found that exponential functions had exponential derivatives. And The Charlie Hypothesis with the Christi Addendum found that cosine was the derivative of sine and that the opposite of sine was the derivative of cosine. This lead to a question about the derivative of the tangent function.
We have proven nothing yet, but it really is amazing when you give students a tool to explore and allow them to use it they will learn things. They don't learn what you were expecting (exactly) or the order in which you wanted to teach it, but that is so cool! They stumble forward and they begin to accept that making a few errors along the way is fine because we are learning. All of a sudden the students are leading the discussion where you (mostly) wanted it to go and the are eager to see if their thoughts are correct.
Together, using Desmos (this is a common tool for me in all my classes) we created a file similar to this where students could plug things in and visualize things instantly.
Because I named the first students hypothesis, every students wanted to share and get their own named hypothesis. If we want students thinking we need to celebrate their thoughts and encourage exploration. Proof can come later after we already know what we are trying to prove. Pretty simple. I need to remember to take the time to do this in all my classes.
In my calculus class, a student noticed that there was a pattern for the tangent slopes of the function \(y=x^2\) namely that the slope was always twice the input. He went on to hypothesize that for the function \(y=x^3\) the slopes would be the absolute value of three times the input. I named it after him (The Christian Hypothesis). [Note: I know the second part in incorrect, but we have a statement we can verify later -MrC]
Today, as we continued to explore slopes and derivatives (the function that finds the slopes of a function) the hypotheses came pouring out. The Mia/Daisy Joint Conjecture found that the derivative of a parabola was a line, the derivative of a cubic was a parabola, and the derivative of a fourth degree function ( \(y=x^4\) ) was a cubic function. The Bre Hypothesis found that exponential functions had exponential derivatives. And The Charlie Hypothesis with the Christi Addendum found that cosine was the derivative of sine and that the opposite of sine was the derivative of cosine. This lead to a question about the derivative of the tangent function.
We have proven nothing yet, but it really is amazing when you give students a tool to explore and allow them to use it they will learn things. They don't learn what you were expecting (exactly) or the order in which you wanted to teach it, but that is so cool! They stumble forward and they begin to accept that making a few errors along the way is fine because we are learning. All of a sudden the students are leading the discussion where you (mostly) wanted it to go and the are eager to see if their thoughts are correct.
Together, using Desmos (this is a common tool for me in all my classes) we created a file similar to this where students could plug things in and visualize things instantly.
Because I named the first students hypothesis, every students wanted to share and get their own named hypothesis. If we want students thinking we need to celebrate their thoughts and encourage exploration. Proof can come later after we already know what we are trying to prove. Pretty simple. I need to remember to take the time to do this in all my classes.
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