beckyl, if you relate this to your trainer that you want to focus on "core strength" and you do not feel the attention is given. Let us know and perhaps we can give more advice.
However, it is more beneficial for your trainer to help you since many of the exercises are too difficult to relate through text.
Again the achievement of the this goal could take months (depending on genetics) to see improvement and many people lose focus after a period of time.
Also remember a key word is "balance". The lower & middle back must be as equally trained as the abdominal muscles. If one is substantially stronger than the other it could have a negative effect on posture as well. The abdominal muscles that play the key role in core strength are not developed by doing the normal situps/crunch type exercises. Although those type of exercises are essential as well for other purposes.
One of the abdominal exercises that I was taught by a doctor for an olympic team was to sit on the edge of a bench with an upright position. Take in a deep breath and let out the air. As you exhale flex/tighten you abs and hold as long as you can. Then repeat several more times.
Now days exercise balls, yoga & pilates have become more popular and each of those incorporate exercises that strengthen the core muscle groups.