The best way to test the cognitive development of an adolescent is to go to a clinical psychologist or neurolpsychologist and have their IQ tested. However, in lieu of that route, there are ways to check the developmental progress of teenagers at home.
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Begin by using your preset list of vocabulary words. Ask your child to define each word to his or her best ability. Give a score of 2 points if the word is clearly defined and 1 point if the definition lacks in-depth understanding. Ideally use 10 to 15 words of varying difficulty.
Test nonverbal problem solving skills by observing your child putting together one of the preselected puzzles. Observe their use of strategy. Record such observations as whether they tend to use trial-and-error versus part-to-whole reasoning, such as completing the border of the puzzle first.
Check your child's working memory using the deck of cards. Start by randomly pulling out two cards and read them aloud to your child and then ask your child to repeat it back to you, number and suit. Gradually work your way up to nine cards at a time. Then, repeat the exercise having your child repeat the selected cards in the reverse order. Record your child's longest correct span for forwards and backwards.
Assess processing speed by timing your child doing a verbal and nonverbal task. For verbal skills, time them doing a moderately difficult crossword puzzle. For nonverbal skills, time them completing a 50- to 75-piece puzzle.