When I started to learn piano about 2 years ago, I practiced scales with my piano teacher. For practicing C Major, she recommended repeating the tonic:
C D E F G A B C C B A G F E D C
When I switched to another teacher, he said I should not repeat the tonic and practice the scale like this:
C D E F G A B C B A G F E D C
Both of them were certain that their way was the right way to do it.
How would you advise a beginner (or even an expert) to practice a scale?
There is no "right" way, though I've never heard of it being repeated myself, or seen it that way in scale books. Most music I've seen involving scales doesn't do that sort of repetition either.
In the end it's about what trains your fingers better. If you have difficulty with the repetition, it may be useful to practise it with the repetition. Likewise, if you have trouble keeping the transtition from ascending to descending smooth (or vice-versa) then practising the other way might help.
If you need a concrete suggestion: Do it without the repeated note.
I agree that there is no "right" way. Those two ways test different things:
When you don't repeat the C, it is testing your ability to move up and down smoothly.
When you do repeat the C, it is testing your ability to hit the same note in succession. It is rhythmically harder to play and less intuitive (for me at least). So perhaps this might pull you out of your comfort zone if you find the former version easier.
There are many other variations of playing a scale which would improve different aspects of your playing. In the end it's all about what you want to get better at.
