How to deal with misapplying mathematical rules? [closed]
I taught gifted students at the elementary level and have had success helping students become less careless. I suspect some of my strategies might help you.
If you make mistakes, you are practicing making that mistake, so now you have to undo practicing getting them correct. I would suggest you isolate the kinds of mistakes you make (eg distributing incorrectly - a=4(c+b) becomes a=4c+b)
Once you have a list of such mistakes, go to Khan Academy and practice the specific skill that you are making careless mistake with. On Khan Academy you can isolate a skill and practice it. You will have to get 5 right in a row, before they consider that you have practiced it. After you have practiced whatever skills you have isolated, you can then take Khan Academy's mastery challenges which you do over several days. They will give you different skills that you have practiced and give you a chance to move up levels until you have mastered the skill. This will ensure that you have really made these "math rules" second nature and will also help you speed up.
I have found in general that students who take tests slowly need more practice. With enough practice they will go faster.
I hope this strategy helps.
Here are some more aspects, which could be helpful:
- Distinguish between understanding and realisation
It's good and necessary to gain a proper understanding for mathematical rules. But to correctly apply them is a different story. To do calculations without mistakes you need a lot of routine. The more you perform these calculations, the more you apply mathematical rules, the better you will perform. So, the message is do it, do it again and again.
- There's more than one way to do it
Typically there is not only a single way to solve a problem. Think about different possibilities to correctly apply the mathematical rules you want to master. This will enhance your creativity and it also enables you to check the solution of your problem from a different point of view. It's a smart way to find mistakes.
- Test and training situations are usually different
When you do exercises at home, don't think too much about timing constraints during tests. Use the private time to apply with care the calculation rules and don't be hasty. This way you can better train doing correct calculations. But sometimes it could also be useful to do some kind of test simulation.
- Have fun and don't worry
It's not helpful for your mathematical development to worry too much about mistakes. Better laugh about them and try to see them as opportunity for improvement. Nobody is perfect and the others also fail sometimes.
- Be confident! :-)
In the interest of not repeating things that have been told to you already, I'll try to offer something else. Be it in mathematics, athletics, music, public speaking, or anything based on performance, you will never reach your best unless you practice under duress. From what I've understood, the issue here is test-taking skills, not math skills.
If you're anything like me, during a test, you get nervous. Your heartrate increases, you start sweating a little bit, and your mind starts racing a bit more than usual. I've had the same experience in all of the above activities: mathematics, athletics, music,and public speaking. The only way you can conquer this is effectively simulating duress and practicing under it. You have to get used to it, or else you'll always perform at $<50$% of your capacity.
A few good methods for practicing are as follows (in order of increasing efficacy):
1) Practice with a timer. That is, if you have an hour long test, time yourself for an hour uninterrupted. For me, this doesn't work, because I know in my head that it doesn't really count for anything.
2) Make bets with friends. If you have friends in your classes, and good material to use as sample tests, put money (or some other bet) on your performance on a practice test. Maybe this isn't ideal, but you'll likely feel some sort of anxiety, and most people like to compete to beat their friends.
3) Practice before you sleep and when you aren't in the state of mind for focusing. If you've been reading silently for a few hours, no doubt you are pretty relaxed and able to focus. If you just get in the door and sit down to practice, you don't get time to collect yourself- just like on a test.
4) Practice beyond what you think is necessary. Many times I have felt ready, but in hindsight realized I wasn't. Many other times, I have spent in excess of $25$ hours studying for an exam I felt I was ready for initially. The end result for me has been that the latter case always results in success- no matter how hard the test or how much I panicked.
5) Learn everything. Provided you have time, don't dismiss things as "probably not on the test." If you don't have adequate time, prioritize.
A few good test-taking tips (which you may/may not have heard) are as follows:
1) Easy questions first. This speaks for itself.
2) If time-pressure is your problem, make your goal to solve all but one problem. Often-times the lack of time pressure makes you work better, and you end up having time for the test after-all. Not to mention, $9/10$ is better than $5/10$ from making mistakes everywhere.
3) Eat some chocolate or something with mild sugar content beforehand. This will make you a little bit more attentive. No coffee immediately before.
4) Sleep enough the night before, always. How much is enough is in the eye of the beholder, for me its $6$ hours, but if you need $8$, sleep for $8$.
5) Time out the problems, allot a roughly even amount of time for each one. If you indeed do the easiest questions first, you'll be ahead of schedule.
Maybe this is nothing new, but the way I read your question, this sounded more important than advice on mathematical mechanics.