Chapter 11 — Mathematics as Knowledge Islands

The previous chapter used addition to explain the formation of one knowledge island.

Mathematics contains many separate knowledge islands.

There may be an island for addition, subtraction, multiplication, division, adding two fractions, solving an equation, calculating a matrix, or using a particular theorem.

Each island is a group of Hard-Linked Tags related to explaining, showing, and using one mathematical concept.

What a Mathematical Island May Contain

The addition island may contain Tags for the meaning of addition, the plus sign, examples such as 1 + 1 = 2, calculation methods, and common mistakes.

It may also contain personal memory Tags, such as learning addition together with Mary.

The subtraction, multiplication, and division islands contain their own meanings, examples, methods, and learning memories. The islands may be related, but each remains a separate group of Tags.

Adding Two Fractions Is a Separate Island

The addition of two fractions can form another knowledge island.

It is not simply a larger version of the island for adding two integers. A person may know how to calculate 2 + 3 without knowing how to calculate 1/2 + 1/3.

The fraction-addition island may contain Tags for:

• examples such as 1/2 + 1/2 = 1
• numerators and denominators
• equivalent fractions and common denominators
• step-by-step knowledge for performing the calculation
• common errors and ways to check the answer

It may contain a correction Tag such as:

1/2 + 1/2 ≠ (1 + 1) / (2 + 2)

This Tag contains knowledge about what not to do.

The island is not any one example, rule, or step. It is the group of related Tags and Hard-Links that explains, shows, and helps the person use fraction addition.

One Island Can Reference Other Islands

The step-by-step content for adding fractions may include operations such as 2 × 3 or 3 + 4.

That content can Reference the multiplication or addition island. Those islands supply the mathematical knowledge needed to perform the operations.

The multiplication and addition islands do not become part of the fraction-addition island. They remain separate islands that can be activated and used when needed.

A new mathematical procedure can therefore form its own island while using knowledge from other islands during its steps.

Several Islands Can Work Together

A matrix calculation may activate the matrix island, multiplication island, addition island, and perhaps the negative-number or fraction island.

The matrix island supplies the main concept and step-by-step knowledge. Other islands supply the operations needed during particular steps.

The person does not need to think about how addition or multiplication was first learned. Those learning memories remain stored, but they are not needed for the current calculation.

Focus Keeps Processing on the Problem

While performing a matrix calculation, the person’s focus is on the matrix, the current operation, and the answer.

The addition island may be active, but the person does not normally think about learning addition with Mary many years earlier.

Later, the person may find a long-lost sticker notebook that Mary gave them. The notebook may Reference the Mary Tag and bring back the memory of learning addition together.

The mathematical knowledge and the learning memory can be related to the same island, but different circumstances bring forward different Tags.

Without enough focus, processing may move from the calculation to other related memories:

addition → Mary → elementary school → sticker notebook → childhood memories

The mind has wandered away from the calculation.

Focus keeps the brain processing the Tags and knowledge islands relevant to the current task.

A Growing Collection of Mathematical Islands

Learning mathematics creates and expands many knowledge islands.

Geometry may contain islands involving shapes, angles, distance, and space. Algebra may contain islands involving variables and equations. Calculus may contain islands involving limits, rates of change, and accumulation.

These islands do not all grow from addition. Each forms through learning its own concepts and uses.

However, a mathematical problem can activate several islands and use them together.

From the STLR perspective, mathematical knowledge is a growing collection of separate but connected knowledge islands. References allow one island to use knowledge from another, while focus keeps the relevant islands working on the current problem.