I usually address my students with the above phrase after handing out marked papers, tests and exams. I then provide time for my students to carefully analyze their errors. I also ask them to keep a running record/journal of the patterns of their errors. Understanding how and where you go wrong will lead to enhanced learning and improved grades - a habit often developed by strong math students. It's not unlike me to develop my next test based on a variety of student errors!
How often have you looked over your marked paper and analyzed your errors? When doing so, how many times have you almost immediately realized exactly where you went wrong and wished that if only you had caught that error prior to submitting your paper to your instructor? Or, if not, how often have you looked closely to see where you went wrong and worked on the problem for the correct solution only to have one of those 'A Ha' moments? 'A Ha' moments or the sudden enlightening moment resulting from the newly discovered understanding of the misconceived error usually means a breakthrough in learning, which often means that you'll rarely repeat that error again.
Instructors of mathematics often look for those moments when they are teaching new concepts in mathematics; those moments result in success. Success from previous errors isn't usually due to the memorization of a rule or pattern or formula, rather, it stems from a deeper understanding of 'why' instead of 'how' the problem was resolved. When we understand the 'whys' behind a mathematical concept rather than the 'hows', we often have a better and deeper understanding of the specific concept.
Symptoms & Underlying Causes of Errors
When reviewing the errors on your papers, it's crucial that you understand the nature of the errors and why you made it (them). I've listed a few things to look for:
- Mechanical errors (transposed number, sloppy mental math, hurried approach, forgotten step, lack of review)
- Application errors (misunderstanding of one or more of the required step(s)
- Knowledge based errors (lack of knowledge of the concept, unfamiliar with terminology)
- Order of Operations (often stems from rote learning as opposed to having a true understanding)
- Incomplete (practice, practice and practice, this leads to having the knowledge more readily available)
Success is Failure Inside Out!
Think like a mathematician and learn from your previous mistakes. In order to do so, I would suggest that you keep a record or journal of the patterns of errors. Mathematics requires a lot of practice, review the concepts that caused you grief from previous tests. Keep all of your marked test papers, this will assist you to prepare for ongoing summative tests. Diagnose problems immediately! When you are struggling with a specific concept, don't wait to get assistance (that's like going to the doctor three days after breaking your arm) get immediate help when you need it, if your tutor or instructor isn't available - take the initiative and go online, post to forums or look for interactive tutorials to guide you through.
Remember, problems can be your friends!