The A1c measures the average glucose level. It tells little about glucose spikes. I used to keep mine around 6.9 when I was taking insulin. But my spikes might have been as low as 50 and highs well over 200. Every time blood glucose spikes above about 140 damage is being done to organs.

I have not had any insulin shots or diabetic medicine for over 3 years now and my last A1c was 5.9. Still to high but better than it was when I was taking 100 units of insulin and two diabetic drugs. Optimal would be something under 5.2 according to the doctors I follow. But the A1c still does not tell the tale of spikes. My range never goes above 150 or below about 120 now. With most readings 125-135 even after meals. I would like to get that fasting number down to around 100 but I must still have some insulin resistance or perhaps permanent damage to my liver or mitochondria. I'm hoping over time as cells get replaced my fasting blood sugar will improve. Time will tell.

For a better understanding (if the doctor understands it) of blood sugar a couple other tests along with the A1c will give more information. Fasting insulin is one and the other is C-peptide.

Edit: if a person is taking insulin shots the fasting insulin test will not have any meaning. The test will not be able to differentiate what is exogenous insulin or insulin created by the pancreas.