There are many ways to explore harmonics and the harmonic series. If this is being done for the first time, then the project below may well be the occasion on which the acoustic properties of sound first burst into meaning for the musician. Certainly, it is a revelation suddenly to become aware of the 'interior' of a sound in this way.
First, repeatedly sound a low piano note. Move a finger along the string until a bell-like pitch is clearly heard. Mark it off in chalk (and measure the length if you want to check the ratios given above). Keep going until the individual nodes become too small to be distinguishable. Now refine this process by playing the open string and attempting to hear the harmonics it contains. Lightly touch the appropriate nodal points and then play the open string again to bring out the effect. The ears will gradually awaken to the pitch contents of the sound. It should be possible to hear the 5th harmonic quite clearly and, with practice, up to the 7th and even beyond.
This can be tried with other instruments. Then, try using the voice. Sing vowel sounds: a-e-i-o-u (pronounced aah, ay, eeh, oh, uh) in a continuous rotation. The word 'why', sung slowly and extending all the vowel sounds, works well (especially in a bathroom!) Sing through the nose somewhat, and gradually awaken the harmonics present in the voice. A buzzing electric toothbrush can also boost the effect.