Mathematics is a human invention that is both a language and a tool. The human being has a tendency to create and invent which is based on law, order, and exactness. This tendency is inherent in human beings, which means it belongs to them universally, and therefore it belongs to the child. That’s why Maria Montessori used to say that math should not be difficult for the child to grasp.

Progression through the Montessori math curriculum is not strictly linear. Instead, Maria Montessori envisioned elementary math as a three-tiered progression. The first tier consists of the numbers to ten, place value, and the four operations. The second tier is dedicated to the memorization of math facts. The third tier is where the children study hierarchy, that is, how the numbers in the decimal system are related and grouped. Initially, they work with numbers from unit to millions. Later they learn the concept of numbers is infinite. The children are free to climb from one tier to another while exploring different concepts of math simultaneously.

Children frequently ask for the biggest problems possible. They also enjoy writing their own BIG problems. The younger children practice using the materials representing whole numbers, fractions and decimals, and through repeated experiences with them, they “discover” algorithms or concepts by themselves or under the guidance of the teacher.

Montessori places great emphasis on the study of geometry, and all the math materials have a geometric aspect. Children study lines, angles, and plane figures, as well as linear and cubic measurement. Later they use boxes of cubes and prisms to cube a binomial or trinomial. Through their studies, the students are able to discover abstract concepts of algebra. They also take great delight in further study of different systems of numeration like those used by ancient civilizations.

Math in the Elementary Years

In the Montessori elementary environment, Math is integrated into cosmic education. Mathematics is the base of the universe. The stars and their courses, the distances they are away from us, and the formulas for chemical elements and substances have a mathematical base. The foundations of buildings, the angles of roof beams, the circumference of the wheel, and the rainbow all have mathematical implications. To be able to explore the universe and all that it contains, one has to use mathematics.

We encourage children to explore the world around them and make mathematical connections. Unlike traditional methods of education, we limit the amount of problems we give children to solve. Instead, we encourage children to make up their own examples and their own problems. In general, we hold the children back when we provide problems for them. But what if the child makes up a problem that he can’t work out? Children learn best by struggling themselves. We give them keys and allow them to use them. One of the child’s psychological characteristics at this stage is for enormous work and the great ability of the intellect. We strive to challenge the intellect and let it work. We let children take on enormous tasks, big problems, and big work because hard work engages the reasoning mind. Workbooks and worksheets are demeaning and insulting to the intellect, they have little place in a Montessori classroom and no place in cosmic education.

The Math materials in the elementary class are concrete so that the mind can manipulate and bring out a pattern. Pattern is often brought out through the work of the hands and through a manipulative activity. The concrete materials are used not so much to teach a process, as they are to offer little doses of help to an understanding of that process. Different children need different doses of help, so we only offer the Montessori materials to the children who need them. Likewise, children are allowed to work on tasks for as long as they need to. The purpose of the concrete Montessori materials is to allow children to discover the mathematical processes and reach abstraction.

Early human beings, from the time they had fire and cooked food, were using math calculations. All that has been built up around math was built up to meet the needs of humans. If we can keep this in the forefront of our thought and help the children realize this, then math cannot be difficult. It’s we, adults, who make it difficult by asking for the abstraction before there’s understanding. In our class, we don’t start with the formula; we start with the concrete sensorial representation. Once the understanding is there, the child can arrive at the formula, the understanding, and the abstraction.

The Math Curriculum

Our program starts with “The Story of Numbers”, which helps children understand the power of mathematics and motivates them to continue exploring numbers and even invent their own!

The following concepts are presented to the children as needed and as appropriate:

- Decimal system
- Linear counting
- Patterns, skip counting
- Comparing, rounding, and estimating values
- Place values, large numbers, and expanded notation
- Operations
- Addition
- Subtraction
- Multiplication
- Division
- Creative problem-solving methods
- Time and Money
- Measurement
- Exponentials as powers
- Multiples, factors, and prime numbers
- Understanding divisibility
- Understanding Fractions
- Fractions operations
- Decimal fractions
- Ratios, proportions, and percentages
- Data, statistics, and probability
- Negative numbers
- Numeric and algebraic equations
- Inequalities
- Squares of numbers and polynomials
- Cubes of numbers and polynomials
- Square roots of numbers and polynomials
- Cube roots of numbers and polynomials
- Exponents and scientific notation
- Different bases
- Roman numerals
- Geometry
- Exploration of shapes
- Lines and angles
- Polygons
- Similarity and equivalents
- Perimeter and area
- Measuring angles and polygons
- Solid figures
- The coordinate plane