I don't recommend singing along with the recording or repeating the exercises on your own. In fact, I don't recommend any technical practicing at all between your lessons. (I myself practiced exercises at home for a long time, against the advice of my teacher Cornelius Reid, and noticed that I made much better progress after I stopped!)
Not practice exercises? Does that sound shocking? (But maybe just a bit liberating as well?) Let’s look into why this advice is actually sound.
First, as hard as it may be to believe in this era of working-hard-to-achieve-your-goals, you don’t need to hammer away at a new coordination for your body to assimilate it. In singing, we are working with micro-adjustments of very small muscles. (We are not like weightlifters at the gym, engaged in building up muscle bulk.) Vocal change, even when it seems huge, comes in tiny increments. After each change, your body needs a chance to rest and to integrate the new coordination. This requires time more than multiple reps.
Second, until you are at a quite advanced stage of your training, you will not be able to sing as well unsupervised as you do in your lessons. You learn best by repeating successes and minimizing incorrect patterns. “Practice makes permanent,” as Mr. Reid used to say. You will move forward in your technique faster when your proportion of correct singing is higher. A little correct singing in your lessons advances you further than lots of less-than-ideal repetitions in the practice room. If you don’t practice, you won’t have a chance to sing incorrectly!
A third reason not to try to repeat exercises from your lessons at home is that this kind of mindless repetition is probably useless and maybe even potentially detrimental. In your lessons, I respond to what I am hearing in your voice at each moment. Because your singing differs from day to day, the exercises I propose will be different too, depending on what is going right and wrong in your vocal coordination at the time. Today’s exercises might be completely inappropriate tomorrow. At some point in your training, you will become attuned to your own voice in the way that I am in your lessons. At that point, you will be able to design your own practice sessions at home. Until then, it is better not to risk reinforcing an inappropriate vocal coordination by doing exercises on your own.
So what should you do between lessons. . .and what should you do with those lesson recordings? Remember that much of vocal training involves developing your ear and strengthening the connection between your mental conception of a tone and your body’s reflexive response to that thought. Listen to your lesson recordings as objectively as possible, noticing the changes in sound and paying attention to what thought processes brought about these changes. See if you are able to detect in your recorded voice the qualities you hear me mention. Let the recordings be a stimulus for thinking, rather than doing.
If you want to practice something, explore vowel sounds. You can speak them or sing them, noticing what you do to form them, seeing if you can do less to form them, finding their nuances and colors, changing quickly or slowly from one to another. Otherwise, study your music. Memorize it, speak the lyrics, analyze it structurally at whatever level you are able. Sing phrases of it in the shower or around your home, and notice what is changing in your technique. A few minutes a day of carefree vocalizing will do no harm. Let it be fun.