in an interview, Zig Engelmann said that he doesn't believe in using manipulatives for teaching math. I thought it was universally accepted that kids start with counting blocks, fingers, etc
Here is what Engelmann and Carnine have written on the topic or the need for making mathematical connections:
Connecting Math Concepts presents skills in applications in a way that permits children to connect concepts.
If the program did an effective job of articulating connections between various concepts, the child who completes the first level would have a strong conceptual foundation on which to build...
How does Connecting Math Concepts forge these basic connections?
The principal vehicle is the number line. The number line is potentially dangerous for teaching number facts because it provides children with a method of figuring out the answer without attending to relationship. To figure out 4 plus 3, the children simply go to 4 on the number line and then move three spaces further. Although children learn the right answer, they don't easily learn the relationship between 4+2 and 4+3 because they have never been required to do anything that focuses on the relationship. The just follow the number line as a "rote" procedure.
A similar problem exists with any method that permits children to count and figure out the answer to problems like 4 plus 2. Finger operations are logical. (Hold up 4 fingers; hold up 2 more; count all.) However, these operations don't prompt systematic learning of relationships because children are not required to attend to relationships. Line-making operations suffer from the same problem. They provide children with a reliable method for figuring out answers to problems; however, they do not induce fact knowledge or knowledge of relationships. Furthermore, they often militate against systematic learning of facts because so long as the child is able to use finger-counting operation or a number line, what motivation is there for learning facts?
Connecting Math Concepts, Level A, Teacher's Guide, pp. 6-7.