The passage from Concrete is abstract thinking is the most important concept in teaching math. Without abstract thinking the student will not be able to grasp most mathematical ideas.
The manipulatives cannot teach the students themselves. A good teacher helps to make abstraction happen. Assumptions by Clements and McMillen.
quesinaire
The following are manipulatives used to teach algebra
We cannot forget about computer software
this is a video of Two students playing a game with dice to practice two digit addition