Basic Operations with Cuisenaire Rods


Hans Ruegg

Soy suizo-peruano, padre educador de dos hijos adolescentes. Vivo en la sierra peruana y he trabajado como maestro teológico, como profesor de inglés, como capacitador de colaboradores en el ministerio con niños, y en un proyecto de refuerzo escolar y educación alternativa. - - Formación profesional: - Certificado suizo de Madurez en Matemática y Ciencias Naturales - Magíster en Teología (Basilea, Suiza) - Diplomado en Educación Cristiana (Antorcha Ministerios Educativos, Perú) - Asesoramiento por la Fundación Moore para educación en casa - Diversos cursos a distancia en las áreas de Matemática, Educación, y las TIC. - - - I am a Swiss-Peruvian, homeschooling father of two teenagers. I live in the Peruvian Highlands and have worked as a teacher of Theology and English, as a trainer of staff for Children's Ministry, and pioneering a project of remedial classes and alternative education (K-12).

Reviews (1)

Ana Molina


A free online course for homeschooling families who wish to improve their children's math learning experience. This is a follow-up course of "Active Mathematics for Homeschooling Families".

This course is also offered in Spanish.

12 weeks (8 hours per week), or at your own pace.

Course Description

This course presents active and creative methods which help children to learn the basic arithmetic operations (addition, subtraction, multiplication and division), and the foundational mathematical concepts related to these operations. Course participants (parents and teachers) will implement these methods with their children. At the same time they will themselves practice mathematical thinking.
The most important goals of the course are that participants:

  • Investigate and understand the mathematical laws related to the four basic operations.
  • Transmit these mathematical concepts to the children they teach, in a way adapted to their level of understanding and of mental development.
  • Become capable of designing and realizing practical mathematical activities with children from about five to ten years.

For whom is this course?

This course is mostly designed for homeschooling families. This surrounding provides the best opportunities and the greatest freedom for practicing the methods suggested in this course. But other families, school teachers, students, and other interested participants may also benefit from this course, adapting the activities to their specific situation.

The activities suggested in this course are optimized for children from 5 to 10 years old. Additionally, the course contains some theory segments and investigation challenges for educators (adults) and older students.

Dates: Free. Participants can study the course at their own pace. It is recommended to invest at least 12 weeks; the activities with the children can last longer.

Course Structure and Workload:

The course is divided into 12 blocks or "weeks". Each block contains suggestions for activities with smaller children ("beginners") and with somewhat older children ("advanced ones"). There are also four blocks with suggestions for preschoolers.
Additionally there are four theory blocks for adults which explain the underlying pedagogical and mathematical principles.

Completing the course requires:

  • Studying each week some short instructional texts and videos.
  • Implementing with the children the suggested mathematical activities. (Choose them according to the children's level of understanding.)
  • Completing 4 quizzes with questions of review and reflection about the course content.
  • Completing 3 educational reports about your experiences in the activities with the children.

Additionally, it is much recommended to participate in the discussion forums related to the course.

The estimated minimum time required is 8 hours per week. About half of this time (4 hours) will be invested into the practical project with the children.

Prerequisites for participants:

A disposition for changing your views about math, and for practicing new ways in learning and teaching math.
Opportunity for realizing the practical projects with your own children or with a comparable group.

It is very much recommended to complete first the Introductory Course of "Active Mathematics for Homeschooling Families", before you study this course.

This course presupposes that participants have already acquired some experience in implementing our method of active mathematics, as presented in the introductory course:

  • Learning math by your own activity, and with concrete operations.
  • Learning math according to the child's developmental level.
  • Learning principle-based mathematics.
  • Learning math by your own investigation.

Also, that participants have acquired the necessary attitudes and abilities for succeeding in an online course; and that they are familiar with using the course platform.

No special mathematical knowledge is necessary.

Materials needed:

  • A sufficient quantity of Cuisenaire rods.
  • For the activities with smaller children ("beginners") it is recommended to buy or make also some bead strings of sizes 1 to 10.

The study materials for the course (texts and videos) are provided free of charge over the internet.

Here you can preview some of the videos from this course.


Summary of Course Contents

For educators:
Review of the educational principles of active mathematics.
Representation of mathematical operations and principles with Cuisenaire rods and with bead strings.
Mathematical principles related to the four basic operations.
Challenges for mathematical investigation, to train your own mathematical thinking.

Activities with preschoolers:
Sorting and classifying Cuisenaire rods.
Constructing mosaics and towers, freely and following instructions.
Symmetry-related activities.
Concept and symbols of "equal", "greater than" and "less than".
First numbers.

Activities with "beginners":
Introduction to addition, subtraction, multiplication and division, in the number space until 20 and until 100.
Measurements on the number line.
Principle of the inverse operation.
Principle of the whole and its parts.
Principle of the product and its factors.
Commutative and associative laws.
Distinction between additive and multiplicative reasoning.
Combining operations.
Divisibility by 2, by 5 and by 10.

Activities with "advanced" children:
Calculating with concrete materials, and mentally, in the number space until 100 and until 1000, practicing the four basic operations.
Principle of the whole and its parts.
Commutative, associative and distributive laws.
Challenges to investigate and reason about mathematical games, and about little-known arithmetical methods.
Factorization of numbers, prime factors, divisors.
Divisibility by 3, by 9, by 4, by 8, and by other composite numbers (6, 12, 15, ...).
Greatest common divisor (gcd) and least common multiple (lcm).

All topics are introduced with practical and playful activities, using Cuisenaire rods and other concrete materials.

Course content

  • Overview of Weeks 1 to 5: Addition and Subtraction

  • How to write your educational report

  • Week 1

  • Week 2

  • Week 3

  • Week 4

  • Week 5

  • Overview of Weeks 6 to 8: Multiplication

  • Week 6

  • Week 7

  • Week 8

  • Overview of Weeks 9 to 12: Division

  • Week 9

  • Week 10

  • Week 11

  • Week 12

Interested? Enroll to this course right now.

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