Completing the Loop
I created this form of practice as a means of accomplishing several things. I wanted students to think about validation and the problem with textbook or instructor provided problems is that the student see them as the ultimate source of validation.
In pedagogy/eduspeak terms I want to return the power to the students; I want to help them understand that there is no mathematics God - certainly not me - no authority that they need go to for a sense of whether what they have done is good or not.
Of course, I can easily look at an algebra student's work and tell them whether what they have done makes logical/mathematical sense, but so can they - so can you! If - and here we have a big if - you understand the rules of mathematics or have a tool
for validation. So to me, validation is a critical piece of the puzzle. Anyhow, I digress. Completing the loop essentially contains several pieces -
- Creating the problem - this can be a tricky step, but with some thought I can generally give concise instructions about how to create a suitable problem especially if we are just talking about a skill, a particular tool in the toolbox.
- Solving the problem - the opportunity to use that particular tool
- Validating the solution - will often depend on previous skills which can be practiced (and provides further validation of the student's ability)
Now, I can see that what I have done, at least potentially, is tripled the amount of work a student has to face - from one step - solving the problem - to three steps. But I would argue that developing this as a tool would lead to greater confidence and understanding, and deeper levels of understanding. Now, I lack the evidence to back that claim up, all I have right now is my intuition and theoretical principles.
Here is an example of a loop for precalculus. In the loop the students would practice finding inverses along with several other subskills and then validate their inverse.
