Mesa College has a good counteragument essay structure example and explanation.
While opponents of my subject make some good points against my position, they are in the larger sense wrong when we consider that they fail to see and interpret correctly ____________, ______________, _______________, and _______________.
While Author X is guilty of several weaknesses as described by her opponents, her agument holds up to close examination in the areas of _________________, ______________, _____________, and ______________.
Even though author X shows weakness in her agument, such as __________ and ____________, she is nevertheless convincing because . . .
While author X makes many compelling points, her overall argument collapses under the weight of __________, ___________, ___________, and ______________.