Once we know what we are counting, the next question is:

How are we counting it?

Mathematics has developed many efficient ways of working with quantities. Sometimes we count one at a time. Sometimes we count in jumps or equal groups. We may combine quantities, separate them, compare them, partition them into equal parts, scale them, transform them into equivalent forms, or examine how they relate to one another.

As students progress through school, the mathematics becomes more sophisticated, but the underlying thinking remains connected. We are still working with quantities; we are simply developing faster, more powerful, and more efficient ways to work with them.

Learning these methods is similar to learning how to move. Crawling, walking, running, riding a bicycle, and driving can all take us from one place to another, but each method allows us to travel farther and accomplish more. In the same way, addition, subtraction, multiplication, division, exponents, formulas, algorithms, and other mathematical methods allow us to work with increasingly large, complex, and abstract quantities. 

Students learn to:

• count one at a time;

• count forward and backward;

• count in jumps or equal groups;

• combine and separate quantities;

• compare quantities and find differences;

• partition quantities into equal parts;

• count how many groups fit into another quantity;

• scale quantities up or down;

• compare rates and ratios;

• count area, volume, and other measurement units;

• count possible outcomes in probability;

• compare change, such as rise and run or distance and time;

• count factors, powers, and algebraic terms;

• and rewrite quantities into equivalent forms when necessary.

Sometimes the counting can happen immediately. At other times, the quantities must first be reorganized or rewritten into compatible forms. In Let’s Count On Math™, we call this making change. Regrouping a ten as ten ones, converting measurements to the same unit, finding equivalent fractions, and factoring an algebraic expression are all ways of changing the representation while preserving the quantity.

The methods students use are often expressed through familiar mathematical tools. Formulas, algorithms, rules, and procedures are not separate ideas; they are efficient ways of carrying out recurring mathematical actions. 

For example:

• • • an addition or subtraction algorithm organizes place-value counting;
  • • a multiplication algorithm organizes the counting of equal groups;
  • • a division algorithm counts how many groups fit, using increasingly efficient chunks;
  • • fraction algorithms organize the counting of equal fractional parts;
  • • area and volume formulas organize the counting of square and cubic units;
  • • the relationship between distance, speed, and time organizes the comparison of three related quantities;
  • • exponent rules organize the counting of repeated factors;
  • • and algebraic rules organize how quantities can be combined, transformed, and compared.

The goal is not to replace formulas, algorithms, or rules. These are powerful mathematical tools. The goal is to make the underlying action visible so students understand what the tool is doing and why it works.

When students recognize how they are counting, a procedure becomes more than a sequence of memorized steps. It becomes a meaningful and efficient way of working with quantities.

Rather than seeing mathematics as hundreds of disconnected methods, students begin to recognize a smaller set of recurring actions that appear again and again across the K–12 curriculum.

Find the counting. Find the meaning.