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Sterilise the jars and any other equipment before you start (see tip). Put a couple of saucers in the freezer, as you’ll need these for testing whether the jam is ready later (or use a sugar thermometer). Put the plums in a preserving pan and add 200ml water. Bring to a simmer, and cook for about 10 mins until the plums are tender but not falling apart. Add the sugar, ground cinnamon and lemon juice, then let the sugar dissolve slowly, without boiling. This will take about 10 mins.
Increase the heat and bring the jam to a full rolling boil. After about 5 mins, spoon a little jam onto a cold saucer. Wait a few seconds, then push the jam with your fingertip. If it wrinkles, the jam is ready. If not, cook for a few mins more and test again, with another cold saucer. If you have a sugar thermometer, it will read 105C when ready.
Take the jam off the heat and add the cinnamon sticks (if using) and the knob of butter. The cinnamon will look pretty in the jars and the butter will disperse any scum. Let the jam cool for 15 mins, which will prevent the lumps of fruit sinking to the bottom of the jars. Ladle into hot jars, seal and leave to cool. Will keep for 1 year in a cool, dark place. Chill once opened.