Wash the quince, wiping off any fuzz. Quarter and core them (carefully, as they're quite hard and dense), then grate into shreds using a food processor with a grating disk attachment, or a box grater. There's no need to peel the quince, as the skins will all get blended into the finished jam.

Combine the grated quince and water in an 8-quart dutch oven or heavy-bottomed soup pot. Cover and bring to a boil over high heat (this takes about 12 minutes), then turn down to medium and continue to cook for 10 minutes. Drain the quince in a large colander, reserving the liquid (a little less than 6 cups) for quince jelly, if desired.

Return the quince pulp to the pot and add the sugar and strawberries. Put the pot back on high heat, stir, and cover, letting it come up to a hard boil (about 10 minutes). Stir again, turn heat down to medium, cover the pot and leave it alone for 10 more minutes. Stir, cover and let cook for 5 minutes. Repeat at five minute intervals until the strawberries have softened and broken down.

Remove the pot from the heat. Use an immersion blender (make sure it has a metal stick, as the mixture is very hot) to puree the jam until smooth. Return the pot to the stove over medium heat, cover, and let cook 5 more minutes. At this point, the jam is like molten lava. It’s going to spurt and spew like the dickens, and you don’t want to get burned.

Turn off the heat, and let the jam sit for five more minutes, covered. Then uncover and stir one more time (wearing long sleeves, oven mitts, and standing back. It still might burble!).

Ladle the jam into hot, clean jam jars. Top the jars with their lids and bands, and water bath process for 10 minutes (or according to the "jam" program on an electric hot water bath canner).

Cool the jars undisturbed for at least 12 hours, then remove the bands, wash the jars, and store for up to one year.