Generally speaking, arithmetic instructional design and student learning should always follow a structured process: tactile, pictorial representation, paper-pencil. As expected, the learning process begins with a hands-on concrete manipulative and transitions to an abstract one.

In engineering and physics, prototypes and physical modelling are often constructed to better understand the actual behavior in relation to the corresponding mathematics. They assist in understanding the behavior and mathematical correlation prior to full-scale production. For example, in the early days of curved highway bridge design in Texas, structural engineers surveyed and plotted the full scale dimensions of a curved bridge structure in a nearby grassy field to verify their paper-pencil mathematical beam chord calculations. The geometry is complicated and since curved bridge geometry was in its infancy, those early structural engineers had no experience in the geometric design of a multimillion dollar curved bridge structure.

*So, what is the point?*

Professionals working in mathematical and science fields may use tactile and manipulative methodology to verify and better understand the abstract paper-pencil math computations. Developmentally speaking, elementary-aged children require the same concrete to abstract learning approach when taught math concepts that they have not seen before.

Students can memorize the abstract paper and pencil mathematics, but the question remains, “Do students understand the physical model that the paper-pencil math calculations represent?”

*An elementary school aged example depicting the process*

A third grade math class is scheduled this week to learn one digit by one digit multiplication. The teacher plans the lessons in the following general sequence.

Core lessons center around single digit multiplication using tactile, physical objects positioned in a set of equal groups representing a typical multiplication fact (i.e. 4 x 5 represented by 4 groups with 5 objects in each group and 5 groups with 4 objects in each group). Lessons are presented over several days with accountability as students write the correct multiplication fact with regard to tactile groupings.

Core lessons transition from tactile manipulative to a pictorial paper representation using number lines illustrating a series of equal jumps (i.e. 4 equal jumps of 5 and 5 equal jumps of 4). Students also draw a pictorial model of the manipulative presented in the earlier tactile lessons of equal numbered objects in each group.

Core lessons focus on paper-pencil computations, but teachers also require their students to draw the pictorial model on a couple of the problems to reinforce physical understanding. Eventually, students memorize their single digit math facts without drawing or needing to draw the pictorial representation - the physical model and the number math are connected.

Pragmatically, the paper-pencil math must be assessed for each child to ensure they possess understanding. A structured and differentiated numeracy program provides student mastery verification for both the pictorial paper representation and paper-pencil learning in a daily assessment. Hence, two of the three learning modes are embedded within the numeracy sequencing, and teachers are assured that previously taught mathematical concepts and skills were retained. Simply put, *Verification of Mastery* of their learning is assessed and confirmed.