GA Flight Instructor with student reported an NMAC with an aircraft traveling in opposite direction near IMS airport.

Narrative

Climbing from ZZZ; birds passed overhead; leaving droppings on our windscreen. Once in level flight about 15 miles SSW of IMS; we noticed Aircraft Y in the distance; seemingly well above us (we were at 2;300-2;400 feet MSL). Because of a combination of hazy conditions and the bird droppings on our windshield; we lost sight of Aircraft Y; when we noticed it; it was less than a mile from us; 100-200 feet above us; traveling opposite direction. We nosed over and to the right in an evasive maneuver as Aircraft Y passed over us.In that situation; we could have stayed in the pattern at ZZZ; returning to clean the windshield before commencing our training flight to IMS. The student and I reacted appropriately once we realized the potential accident developing; but if we had been able to maintain visual on the opposing aircraft; we could have maneuvered to evade much sooner. The student and I will revisit the FAA Safety Team literature/presentations on aircraft safety in flight and VFR collisions in 'practice areas' (as we were crossing the ZZZ/IMS/LOU practice area when the near-miss occurred).