You will note that I continually state that the purpose of using manipulatives is to teach concepts. Manipulatives are used to give the students a hands-on experience, one that is memorable and helps them interact with different representations. This is a core principle of UDL (Universal Design for Learning).
The goal of using manipulatives is to illustrate a concept and then get rid of them. Students should seldom perform calculations with manipulatives unless it is skill building and aids in memory. For example, using manipulatives to illustrate/see calculations of large quantities using craft sticks and a place value mat is extremely useful...for a while. It reinforces our place value system and allows them to physically experience regrouping and renaming-a concept that is one of our "continuous threads." Once the student begins to recognize the concept and has formed a mental representation of the procedure involved, we would want to move the student to the representation and abstract levels.
Picturing groups of quantities can certainly explicate the meaning of multiplication and division. They can help automatize select facts. They can illustrate the concepts of multiplication and division easily. They should lead to the use of specific fact families-and for LD students a very few- which are practiced and applied to the automatic level.
An inefficient use of manipulatives would be using counters to solve successive problems beyond the child's fact base knowledge system. This is where the general education teacher and the special educator may part ways in using a book or set curriculum. The published curriculum assumes that the child using the textbook has attained certain skill levels. The special needs student may not have the skills required to use the worksheets and practice pages associated with a specific concept.
This is not to say that a special education student cannot be taught higher level concepts. It only means that, as the What Works Clearinghouse suggests, struggling students practice math facts daily and as I say, use THOSE facts in their activities. The teacher may make up a worksheet...yes, in your spare time of course...to fit the needs of the struggling student. Using a computer program such as Math Type, or the equation editor in MS Word, the teacher can create a simple worksheet with fewer problems on a page, ample white space, and a restricted set of number facts which can be practiced to complex levels.
Take for example, long division. The typical text book would ask that the student work with a single digit divisor and two digit dividends until all multiplication facts have been worked through the division algorithm. Then, as the student approaches multi-digit dividends, the student is expected to have mastered the times table facts. This would preclude the special needs student from doing the activities. It is like learning the entire multiplication tables yet again which plays to the special needs students' weaknesses.
The special educator can introduce the concept of long division using Unifix cubes. By choosing a quantity such as sixteen and asking the student to make groups of various quantities ( 3, 4, 7,) the can experience one of the meanings of division including with "left overs" or remainders. Then the student can learn the division algorithm- the goal of the lesson- using friendly numbers. The special education teacher can easily create a worksheet using one of the tools mentioned above. The worksheet might use only a single times table throughout but include problems with varying levels of complexity. An example might be 30 ÷ 5, then 32 ÷ 5, then 34 ÷ 5. Eventually the algorithm might be extended to 350 ÷ 5 and 360 ÷ 5. The student would encounter the procedures in a meaningful way and learn the steps to the traditional algorithm incrementally, sequentially and meaningfully. This would prevent a student from simply using counters to solve problems by hand repeatedly practicing unrelated and isolated facts independently of each other, and leading to frustration without serving to build any mastery of any fact families.
As always, look at what you are teaching. Decide what your goal is. Are you teaching a concept or practicing applications? Are you focusing on procedural fluency or torturing a student with automatic recall of facts while trying to teach sequential steps.
Remember to separate the concept from the computations and build fluency while maintaining skills. Applications do involve complexity, but students should be moved from using know facts to incorporating new ones in ways that build fluency and competence and do not lead to frustration.