This concepts-first curriculum challenges popular beliefs that students could not adequately understand concepts withour prior, or at least simultaneous, mastery of skills. Having seen the principle of defining a sample element to attain the area under a curve, the transfer of this knowledge to find volume illustrates a student built schemata. Find the volume of the solid created by rotating the enclosed area from the previous example around the x-axis. This, for many students is difficult to visualize. Unfortunately, the algorithmic apporach will generate the correct answer leaving the student with no understanding or image of the process.
Students can now place a sample disc or very thin cyclinder, in the 'horn'.
This disc will have a radius of y and a height of dx. The volume formula for a disc is applied and the volume is attained by using the integral function on the TI-85 . A similar procedure will show the rotation around the y-axis. Once again giving the students the opportunity to 'see' the shapes. As the rotations become more complicated the students start working collabortively, questioning each other's approaches and defending their own solutions.