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Typically, it is more effective to start with solving for unknowns and end by solving for systems of variables and numerals. Graphical plots of the answers may help students understand that multiple answers are possible.

Concrete objects do not teach students, teachers do. When using concrete objects, students need to be explicitly taught why and how to multiply, divide, add, subtract, factor, and so on. The concrete objects provide an additional means by which to understand the procedures and reasons for calculations. For example, using concrete objects, students have to understand the operation being calculated in order to know what to at for the answer. For addition, the total number of onesticks remaining is the answer. For subtraction, the total number of onesticks remaining is the answer. For multiplication, the total number of onesticks in all the cups is the answer. Multiplying (3) (5) means three groups of five objects. The result is three cups of 5 onesticks each. The total number of onesticks is 15. For division, the number of onesticks in a single cup is the answer. Division requires students to split objects into parts. The total per part is the answer. 8 onesticks divided into 2 cups means four onesticks per cup. Each cup has four.

Representational learning is the most useful learning steps for students to acquire. Students take advantage of pictorial representations when faced with abstract tests where concrete materials are unavailable. Also, pictorial representations are more useful than concrete learning when there are is a lot of notation or large numbers in the problem. In these cases, concrete manipulations may be cumbersome to answer the problem. Representational learning, however, may be more manageable.

Representational learning must relate directly to the concrete lessons. It is important to use the same terminology and discourse with the students during representational lessons as concrete lessons. For example, still call drawings of sticks onesticks and call circles or cups as cups or groups. This helps students relate the pictorial representations to the concrete objects.