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.


” 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

Mathematics are beautiful. Yet schools and curriculum programs beat out this beauty through worksheets, boring lessons, and memorizing formulas. Formulas and worksheets are not math; no more than the dots and lines on paper are music. The formulas and symbols on paper represent an mathematical idea, but they are not mathematics. We teach children the symbols so they can communicate their ideas to others. They should be discovering ideas with our guidance; not sitting passively listening to us preach.

Math is about asking questions, discovering patterns, and solving problems. Give your child free-reign of manipulatives, and ample problems to think about problems. Ask questions like: How would you solve this? What do you notice? What does this remind you of? How did you figure that out? Is there a pattern? Will this always be true? If not, when is it true and when is it false? In her book, Let’s Play Math, Denise Gaskins lists three things that mathematicians do: avoid busywork, ask questions, and love to play. When looking for a math curriculum (or preparing math activities) for your child avoid anything that is full of direct-instruction, busywork, and no games. 

Today, American students rank 36 out of 79 countries around the world in mathematics performance. Over time, math instruction has moved from nurturing the God-given love of patterns and allowing children to discover those ideas on their own, to drilling formulas, filling out endless worksheets, and high-stakes testing. 

In Arithmetic for Parents, Ron Ahroni explains that math should be learned starting concretely, then verbal, and finally to abstract. For example, you notice your five year old counting her crackers. You create a story problem by asking “Let’s pretend you gave me three of your crackers, how many would you have left?” Keep playing with objects in all different combinations until she has solidified the idea in her mind.
When she has a solid foundation of adding/subtracting with concrete objects, you can play with story problems, each of you taking turns verbally expressing your mathematic idea as a game. Finally, in formal lessons, you teach your child how to communicate her story problems and ideas using symbols on paper. Symbols on paper are the last step and should only be taught once your child understands and is familiar with the idea concretely. This applies to all mathematics, even algebra! 

One of the best ways to ignite excitement and interest in your child’s mind is to read mathematician biographies. Biographies include the questions, problems, and triumphs the persons faced before discovering a mathematical idea. My own children love biographies and better see the beauty in math.

For more detailed instruction on teaching each of these subjects across the forms, download the Curriculum Guides available in “Download