Integers, Fractions, and Measurement in Primary Education

Posted on Mar 24, 2025 in Other university degrees and diplomas

Integers in School

The study of integers is justified in primary education due to their presence in everyday life (temperatures, bank accounts, elevators…).

Difficulties Students Face Manipulating Signed Numbers

Working with signed numbers means breaking with the traditional understanding of numbers as simply representing a measurement or absolute magnitude. In this context, zero indicates the absence of magnitude, making the concept of numbers less than zero challenging. Addition is associated with increasing or adding, so the result must be greater than zero, or at most equal to the summands. Subtraction is associated with removing or separating, so the result must be less than zero, or at most equal to the minuend.

These statements are inherently linked to the concept of numbers and have a decisive influence on its construction. Accepting the existence of signed numbers requires a restructuring of the concept of number, as they do not generally have the properties described above.

Obstacles Students Face

  1. Leaving the plane of reality to conceive abstract concepts.
  2. Considering addition only as an increase.
  3. Interpreting the order of negative numbers as if they were natural numbers.
  4. Forgetting the signs too often.

Fractions

A fractional number, or fraction, is a pair of integers, written in the form a/b, which is used in different contexts or situations. Initially, these contexts may seem unrelated, but they share a common underlying concept.

Classes of Fractions

  1. Part-whole
  2. Comparing a subset of items and a set
  3. A point on the number line
  4. The result of a division
  5. A method of comparing situations or sizes of two joint measures

Fraction as Part-Whole

The fraction indicates the ratio between the number of parts and the total number of parts into which the whole has been divided.

Criteria
  1. A region or area is deemed severable.
  2. The whole can be divided into the required number of parts.
  3. The parts or divisions are exhaustive and cover everything.
  4. The number of parts and the number of cuts need not be coincident.
  5. Pieces or parts must be the same size.
  6. The parts may be considered like everyone else.
  7. The whole is preserved.
Difficulties
  • Understanding of unit fractions.
  • Understanding that subdivisions are equivalent.
  • The transition between the verbal (three fifths) and symbolization.
  • Identification of a unit in a diagram showing more than one unit.

Fraction as Points on the Number Line

If we have the fraction a/b, each unit segment is divided into b equal parts, of which a are taken.

Equivalent Fractions

Equivalent fractions are obtained by multiplying the numerator and denominator by the same number.

Activities
  • Exercises to complete the missing number in equal fractions.
  • Compare and order fractions with the same denominator.
  • Compare and order fractions with the same numerator.
  • Problems.

Magnitudes and Measurement

A magnitude is the physical property of a body that can be measured. To establish the dimensions of a magnitude, it is enough to compare it with another reference that we call unity. The International System of Units is currently used by the scientific community.

Measurement is an action that children cannot perform easily, making it almost impossible to practice measurement until late elementary school. This difficulty arises because the act of measuring requires significant experience.

Stages That Must Be Overcome

  1. Consideration and perception of a magnitude.
  2. Preservation of a magnitude.
  3. Ordering with regard to a given magnitude.
  4. Relationship between size and number.

All these stages will be achieved if the child reaches mental maturity.