Does it sound strange to use "hands-on background"?

Solution 1:

Lexico (The Oxford Dictionaries) gives the following definition

hands-on
PHRASE

1 Involving or offering active participation rather than theory.

The time is equally divided between interactive theory classes and hands-on practical classes.

So it means being involved physically. A thought-based process can't be "hands-on" — that's going back to theory. Another example phrase

As managing director, Rona takes an active, hands-on role in all her company's activities.

means Rona is involved in the actual tasks done by the staff.

Solution 2:

Theory and hands on are not in negation. One is the subject matter, and the other is the approach to it. The term hands-on can be used as an adjective to the approach taken. For example, one may say that Such and such book provides a hands-on background to the theory through plenty of examples. Here it does not sound strange. I would not want the phrase "hands-on" to be the sole property of non-theory topics! A theory can also have an hands-on approach, which indicates learning the theory through its practice. That practice can be through paper and pen, computer simulations, or through other means of laboratory or observations of natural phenomena. So, indeed, there is a plenty of hands-on that goes on in theory too.

A few examples of the usage of the term hands-on in theory topics/maths are as follows:

https://files.eric.ed.gov/fulltext/EJ1086006.pdf

http://www.math.ucsd.edu/~ronspubs/pro_10_hands_on.pdf

https://www.naesp.org/sites/default/files/resources/2/Principal/2004/N-Dp28.pdf

https://learn.org/articles/New_Hands-on_Approach_for_Math_Education.html