"Detail oriented" and "pays attention to detail" are requirements you see on nearly every job posting. This makes sense, since being observant about details means you can avoid costly mistakes. It's a priority for employers and an easy hurdle for people who are already detailed thinkers, but it's more of a challenge for people who aren't naturally detail-oriented. Fortunately, you can work through a number of exercises that can help you build your ability to see and remember details.
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You might remember this game from grade school. Flip over a field of pairs of cards on a table. Turn two over for a few seconds, then return them to face down. Any time you turn over two matching cards, discard those cards from the field. As your memory for the details of where the cards are located improves, so will your ability to perceive and remember details.
Developing an eye for details is mostly about building the habit of looking closely at your surroundings. "What's wrong with this picture?" puzzles and puzzles where you look for small differences in similar pictures are good drills for doing exactly that. As you practice finding the problems, you will begin to develop the habit of finding the small things that make a big difference.
This works similar to memory card games but on a more challenging level. Look at a picture or a scene for one minute, then turn away. Sketch the scene from memory or make a list of everything you can remember from the scene. Once you're finished, compare what you've put down to the reality and see how much you observed. You can scale this by starting with simple line drawings and moving your way up to pictures of street scenes.
Math is an inherently detail-oriented activity. If you get every detail right, it works. If you get a single detail wrong, it does not. Although spreadsheet programs and calculators can do a lot of your basic arithmetic for you, simply doing your checkbook or budget by hand from time to time can exercise the "mental muscle" responsible for being observant.
Bonus Skill: Get Organized
Getting organized doesn't directly affect your natural level of attention to detail. However, it does remove a lot of physical, emotional and mental distractions from your environment. When your desk is clean -- literally or figuratively -- more of your brain is available to notice small details. Exercising your organization leads to a state where you are better able to pay attention to detail.