College Math Teaching

September 25, 2010

The Peril of Teaching Advanced Techniques to Intellectually Immature Students

Filed under: derivatives, differential equations, mathematics education, student learning — collegemathteaching @ 9:41 pm

Ok, I told my differential equations class to solve:
$y' = y(y-2)$ with initial condition $y(0) = 1$
This is straight forward right? I expected the “good” student to separate the variables:
$dy/(y(y-2)) = dt$ and integrate; the left hand side can be done by an easy integration; one can use partial fractions but I allow them to use calculators that have symbolic calculus operations on it.

Now of course, you had the bad student mistake: integrating the right hand side with respect to $y$:
$(1/3)y^3 -y^2 + C = y$ but you will always have some clueless students.

But what surprised me was this: some students used the Bernoulli change of variable method:
$y' + 2y =y^2$ which leads to $y^{-2}y' + 2y^{-1}=1$ and then setting $z = y^{-1}$ which leads to: $z' = -y^{-2}y'$
and that gives $z' + 2z = 1$ which has solution $z = 1/2 + ke^{-2t}$ and then $y = 2/(1 + ke^{-2t})$
Those who did this got it mostly right, though most made algebraic errors along the way.

So I am glad that they knew the more advanced technique but why use it here? 🙂