How can you measure the plant pieces to determine the rate of osmosis?

By measuring the plant's pieces mass, or the surface area to volume ratio.

By measuring the change in mass from initial to after solution indicates the net flow of water, if it increases it proves that osmosis occurred.
If the surface area to volume ratio decreases that also means that osmosis has occurred.

This measurements can’t be used to determine the rate of osmosis but can  prove that it occurred.