Preheat oven to 250 degrees. Put a plate in the freezer. With a potato masher or spoon, crush 1 cup berries in a medium heavy-bottom nonreactive saucepan. Add remaining 3 cups berries and 1/2 cup water; bring to a simmer over medium heat. Cook, stirring occasionally with a wooden spoon to break up berries, until berries are soft, about 6 minutes. Meanwhile, spread sugar out on a rimmed baking sheet. Heat in oven until warm, about 5 minutes. Slowly stir sugar into saucepan; return mixture to a simmer, stirring constantly. Reduce heat to medium-low; cook at a slow boil until thickened, 45 minutes to 1 hour. To test for doneness, drop a small amount of jam onto the chilled plate; it should wrinkle when pressed with your finger.