Organic matter is anything that decomposes and adds humus to your soil, like compost, animal manure, peat, lawn clippings, straw, wood chips, cardboard, leaf litter, and sawdust.
Although sand is ultra-light, mixing it into pure clay can actually make it worse. Instead of lightening heavy clay, sand gives it a concrete-like consistency.
In a sloped garden, create swales or terraces to improve clay soil. Level out areas and create terraces or “steps" to slow water runoff and retain sediment.
Alternatively, build swales (shallow trenches that stall and trap water) to prevent silt and organic matter from washing away, creating a layer of humus and loam over the clay.
If your clay soil is compacted, physically loosen up the soil with a garden fork or lawn coring aerator, but be gentle as over-tilling may actually compact it more.
Instead of tilling, add mulch over the top of the clay soil without digging it in. Over time, roots, insects, earthworms, and water will help the layers mix, loosening the clay.
The more you manipulate saturated clay soil, the denser, more compacted, and less aerated it will become — plus, the clods will just stick to your tools.