General Algorithm Power is the Mathematical King
Students must memorize certain basic mathematics information. It cannot be avoided. But, elementary teachers must understand what kids should commit to memory and what they should not. If teachers separated math into a basic factual knowledge versus procedural knowledge, I believe this distinction would assist in determining if a math topic is a memorization criteria or not.
Factual Knowledge – Commit to Memory
An abbreviated (i.e. not complete) list of factual mathematics knowledge kids should memorize include: four operations in math facts (i.e. addition, subtraction, multiplication and division), basic dimensional facts – inches in a foot, meters in a kilometer, etc. and math vocabulary words (i.e. sum, difference, product, quotient, proper fraction, mixed number, etc.). Of course, it is assumed that when dimensional quantities and math vocabulary words are committed to memory that there is a mental understanding of the magnitude of dimension (i.e. about how big is an inch or a meter) and what the definition of the math term in context to a numerical reference. In short, these are typical elementary aspects that must be committed to memory for automatic recall. Finally, teachers should rigorously press each memory task until each student demonstrates mastery.
Procedural Knowledge – Commit the General Algorithm
with Physical Understanding to Memory
Procedural knowledge, in this case, is a repeatable arithmetic method to compute a solution using basic factual knowledge. It is beneficial in pedagogy to understand the procedural power of a general standard mathematics algorithm – strongly emphasizing the word, “general” used consistently across grade level learning.
For example, when a child learns multiplication beginning in second or third grade, they developmentally and appropriately learn using tactile methodologies to understand conceptual multiplication as either a repetitive adding model or group model. Hence, in the repetitive adding methodology, 4 x 3 may be viewed as either adding 4, 3 times or adding 3, 4 times. Or, the group model – same meaning, of course – the child learns that 3 groups with 4 items in each group equals 12 total items or 4 groups with 3 items in each group is also 12 total items. The child draws a pictorial of the solution after transitioning from the tactile manipulative.
As second semester 2nd graders mature to intermediate students, multiplication computations transition from single digit (i.e. 3 x 4) multiplication to multiple digit (i.e. 16 x 23) multiplication computations. But, most importantly, the physical or pictorial model whether computing single or multiple digit multiplication does NOT change – which is key to student understanding!
For example, when the student is computing the product of 16 x 23, the student must understand that the pictorial model is exactly the same as it is in single digit multiplication. 16 groups with 23 items in each group equals a total of 368 items or vice versa – 23 groups with 16 items in each group equals a total of 368 items.
It is highly recommended to use the standard multiplication algorithm (as shown above) to fully utilize learned math fact knowledge. The algorithm can easily be separated and broken apart so the individual computation components of the algorithm are readily understood. Hence, the same pictorial model representing the physical meaning of the mathematics students learned with single digit multiplication – must be drawn to represent multi-digit multiplication. In that way, students understand ALL whole number multiplication has the same physical meaning – and no mystery is occurring.
There is also the advantage of relating the multi-digit multiplication with the same multi-digit numbers in a division computations. The final pictorial model is exactly the same with the exception that the student is either computing a product in multiplication or the number of equal groups in a long division procedural process. This fact further illustrates the rationale and advantage of teaching a structured general algorithm that is consistent whether computing products or quotients.
When an elementary teacher changes the procedural model to the ‘lattice method’ or elects to use the ‘big seven’ method when teaching students long division, the elementary student does not directly connect with a pictorial and physical understanding of the mathematics as was learned in previous grades for simple multiplication or division computations. The child is learning and memorizing different procedural knowledge methods for the same general whole number operation as well as losing basic place value significance. Unfortunately, the singularity of methodology in arithmetic mathematics is lost by instructional design.
Normally, as a principal, I am open to different teaching methods with the condition that they are effective, but this is not that situation. I did not and I would not advocate as a principal and the instructional leader of a campus for elementary teachers to instruct students to memorize different procedural knowledge models in multi digit addition, subtraction, multiplication or division. It is poor pedagogy in my professional view of arithmetic mathematics.