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currently I'm a math student. I was wondering, how math excercises are designed for books? generally speaking. I get that excercises start easy and gradually increase difficulty but difficulty can get out of hand quickly.
How does /sci/ create their excercises to teach math?
How to design math excercises that gradually increase difficulty?
How to make sure an excercise does not get to complex for writing on the board when teaching?
When writing the firs excercise of a set, sohuld I begin with the most basic and simple excercise or should I get the most complex and strip it down to more easier excercises?
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It's 2026 so most textbook writers grew up with books of integrals, literal textbooks of just integrals, and they pick ones they like or otherwise find enjoyable along with a few difficult ones for the end of the section. Exercises are generally designed to cover all use cases for a specific section/thought/concept, and the latter questions are the most difficult and niche cases to deter students from just mindlessly doing them all without thinking.
What I'm saying is, go look at an old fashioned Tables Of Integrals textbook. It's just 200+ pages of solved integrals, without solutions but with proven solutions somewhere. Then start solving them. If you solve them all you can then assemble any math course up to Calc II based on your own knowledge and your question will be answered.
This is enormously difficult and is what the putnam questions are based on. Many professors have had their entire careers just doing this.
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>>16903254
This is very very difficult task. And if you google "mathematical problem posing", you'll just get mathematics pedagogy nonsense.
But like everything, either you steal it from someone else or you come up with something yourself.
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>>16903254
As others have said, most of these questions have been lifted from prior works over the course of literal centuries.
If you want to make something new, you could try starting with the solution and run the inverse operations of what you intend your students to be using until you have some expression you might expect to have to evaluate in school.
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>>16903254
>Think about various statements which are known to be true and weren't proved in class
>Think about various statements which can be proved using what was covered in class
>Think of examples
>Think of interesting examples
>Think of counterexamples
>Think of examples which might involve certain key techniques
>Think of examples where there is a straightforward, but long and tedious approach, and an alternative elegant and quick approach which requires some thought
Usually when I'm writing notes, these kinds of things will all come to me, and then I have a plethora of exercises I can assign.