Of all his early studies, perhaps none is more important to the child as a means of education than that of arithmetic…The chief value of arithmetic, like that of the higher mathematics, lies in the training it affords the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders” 
Charlotte Mason, Home Education, p 254

“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.”
-G.H. Hardy

“Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.” 
-Paul Erdos


Children learn basic addition, subtraction, multiplication and division using real objects in real life situations. Parents make lots of counting materials available and allow children the freedom to play with them.
Sloyd is a handicraft; cutting and paper folding to create objects, much like origami. In the process of cutting and folding paper shapes, children are introduced (and become familiar) with geometry! A fun and important introduction to geometry.
Out-of-door geography and mapmaking is another concrete method of learning geometry. Children learn how to create a map of their house by finding lengh x width and scaling that down on graph paper, and so many other skills!
Practical geometry builds on geography and sloyd with more difficult, hypothetical questions. Preparing children for abstract geometry problems later on.
Advanced multiplication/division, fractions, decimals, etc. are all taught in arithmetic. Students learn how to communicate mathematical ideas with symbols, like bar graphs.
Your child may have already been introduced to a basic, concrete understanding of algebra through cooking or other activities. Remember, concrete before abstract! But they need to learn how to communicate those ideas on paper using formulas.
By the time students have reached geometry they should have a solid foundation in concrete geometry and will be able to understand abstract problems. Geometry will include graphing shapes, finding area of irregular shapes, etc.
The main reason in learning how to do algebra is not that each of us will be faced with algebraic problems in our life, but the skills we learn and the brainpower we strengthen as we learn how to solve these difficult problems.
Advanced math topics include: calculus, trigonometry, statistics and probability.

Download the curriculum guides for detailed instructions, booklists, and materials for each developmental stage. 


Introduce three to four ideas each term. 

  • Read about concept  in Arithmetic for Parents, or chosen curriculum
  • Pick an idea to present to your child for her to ponder or generate solutions. For example: 
    • A mathematical idea
    • An interesting problem
    • The history of a mathematician (see booklist)
    • The history of the concept (see booklist)
    • A shorter way of solving a problem
  • Prepare an activity or problem to present to child

Introduce New Concept (Generation)

The first five minutes are spent learning about a new concept, or practicing the current one. Present the new idea by giving your child a problem (with manipulatives) and allow your child time to generate a solution before giving guidance. Let them experiment, ask questions, and discover on their own. Use the question examples above to help them see patterns and discover solutions, if needed.

Manipulatives (Concrete)

“A first grader should count as much as possible. This is the only way to establish the concept of numbers. A first-grade classroom should be full to the brim with buttons, beads, popsicle sticks, straws.” (Ron Ahroni, Arithmetic for Parents)

The next ten minutes are spent playing, practicing, and counting. Children first count and “act” out the problem using manipulatives. The next step is to draw a problem on paper, and finally create math stories (word problems) using the new idea. Working with real objects allows children to see patterns and play with math in a way that is not possible with mental math or pure numbers. The majority of time spent in the first year or two of math is counting and working through problems with objects. When children have a solid understanding of numbers concretely, they move to the mental, or drawing, stage. 

Drawing (Mental)

“If you stay with meaningful mental arithmetic longer, you will find that your child, if she is average, can do problems much more advanced than the level listed for her grade. You will find she likes arithmetic more, And when she does get to abstractions, she will understand them better.” (Denise Gaskins, Let’s Play Math)
Once a child has counted and worked addition/subtraction problems with objects, they are ready for mental math and story problems. Some children may want to draw out their story problems, while others just tell it orally.  

Pure Number (Abstraction)

Once children are familiar with concrete and mental math problems. they can finally move on to doing a few pure number problems in their math notebook. The last five minutes is spent reviewing past concepts in pure numbers (abstract form) and writing a few problems in their math notebook. Pure number simply means the symbol for a number removed from any denominators. For example, a mental math problem would be “4 apples plus 3 apples equals_____.” Pure number problem would be 4+3=___. 


In learning sciences this is also called “retrieval practice” and “reflection.”  It is extremely important to regularly review a variety of math concepts, especially as story problems. It’s also important to practice “reflection” by asking questions – What happened? What did I do? How did it work out? What would I do differently next time?