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Maybe it is the long day speaking, but programming is a highly advanced intellectual activity dealing with complex, poorly understood and unpredictable inputs and outputs. Depending on your child’s age, can they grasp the level abstraction we are talking about? It takes many us in this career a lifetime of actual working with these concepts to master this skill and it all gets thrown out the door once you add people to the equation.
Modern programming languages were created to abstract the union of dissimilar technologies and allow us to reduce the complexity to something the human mind can comprehend and, most importantly, influence so we can solve real problems. I commend you for trying to encourage your child to learn how to think in this paradigm, but an old-fashioned language like BASIC will not equip your child with relevant skills or knowledge used in modern software development – the most important skills are people skills, not how great a coder you are. It is far more important to know how to communicate your ideas, influence others, build trust, provide useful feedback and listen to your co-workers, peers and customers.
I would consider the software concepts inherent in BASIC, i.e. structured programming, to be nearly 20 to 30 years old and were made mostly obsolete with the object revolution of the early 1990's (which has not been completely accepted be all in our community). IMEO, asking for a tool that is outdated and inappropriate to teach a way of thinking which is not relevant is like asking your child to learn how to use a slide rule and sine-cosine charts to calculate the hypotenuse of a triangle when other children are using electronic calculators. Sure you can do it the old-fashioned way by hand, but do you really gain anything beyond an esoteric notion that this is really a pain in the posterior?