Math is called a foundational subject for good reason: if you don’t have a solid foundation, anything you try to build on top of it is in danger of falling apart. Math is also called a sequential subject, meaning that math skills must be mastered in sequence, each skill building on the skills before it. This picture represents my view of math skills and the order in which they should be mastered, starting at the bottom and building up, one skill upon another.
No one starts teaching math by instructing their preschoolers in differential calculus. The first math skill we teach is Sorting: Which ones match? Is this one like that one? We may start the sorting process with colors or shapes, but Sorting is still the basic skill being learned. Sorting is the #1 most important math skill, used from recognizing number value to solving the most complex equations. Counting is an extension of sorting, assigning a number name to each different quantity. We “know our numbers” when we can group the correct quantity of pieces to represent any given number. We have mastered counting when we can recite the quantities in ascending order. The ability to count backwards is preparation for further skills yet to come.
Place Value might be considered to be an extension of Sorting by placing 1-digit numbers together in one group, 2-digit numbers as another group, yet another with 3-digit numbers, 4-digits, and so on. Children who are learning to count past 10 are learning place value, even though they are not yet adding or subtracting large enough quantities to require carrying or borrowing. Those skills work hand-in-hand with addition and subtraction, but an understanding of place value has to come first. Using a large quantity of identical small manipulatives, such as toothpicks, you can demonstrate the quantities represented by numbers in the ones column and numbers in the tens column to show how and why we write numbers the way we do. As your student gains skill with addition, you can revisit Place Value to demonstrate carrying into the tens, hundreds, thousands, and as many columns as your child wishes to add.
The next natural step after Place Value is Addition. Your child may already be using his counting skills to inform you that since he already has 1 cookie, if you would just give him 2 more cookies, then he would have 3 cookies! He may not recognize 1+2=3 on paper, but he certainly understands cookie quantities! Addition facts are best learned through using real-life objects, manipulatives, or even diagrams, rather than just expecting a young mathematician to transfer immediately to written problems. Hands-on practice makes subtraction easily evident as the un-doing process for addition, thereby taking away the stigma that subtraction is yet another new skill to learn. If a student knows addition facts to the point of quick recall, that same student will be able to perform subtraction. Therefore, a student who struggles with subtraction is a student who has not mastered addition facts.
Multiplication is often presented as one more new skill to master, but when presented as a “short-cut” to repeated addition, the student will see multiplication facts as a convenient tool, not as an obstacle to further learning. Multiplication facts can be demonstrated with a large quantity of small manipulatives that can be grouped into repeated rows (½” squares of heavy paper or cardboard work very well). Some quantities of manipulatives can be rearranged to show various factors which result in the same amount, such as 1×12, 2×6, 3×4, 4×3, 6×2, and 12×1. Grouping and regrouping the manipulatives will give your student a deeper understanding of multiplication facts as he sees the groups (visual), arranges them with his own fingers (tactile), and repeats the facts aloud (auditory). A kinesthetic learner will prefer standing or kneeling to do this activity, providing yet another sensory element.
Why isn’t Division listed in these Building Blocks? Simply because Division is un-doing Multiplication, just as Subtraction is the un-doing of Addition. The only tricky part to Division is that sometimes things don’t come out completely even, and we get “left-overs”—but every child who has tried to share 5 cookies with 3 friends understands that concept already. Division uses the quick recall skills for multiplication facts to regroup as evenly as possible, and the “left-overs” will be dealt with in more detail later on as these skills progress even further into the concepts called fractions and decimals. By the way, fractions, decimals, and percents are all “nicknames” for the same amounts—they are just different ways of looking at the same quantities, such as ½, .5, and 50%, and those all mean that you and I are sharing equal amounts of the same cookie!
The final Math Building Block to be mastered is Logic. Logic means making sense of things, so they come out right. Logic may come in the form of “If/Then” statements, such as the block in the picture shows: If all cats have 4 legs, and Fido has 4 legs, does that then mean that Fido must be a cat? Fido might be a cat, but we also know that other animals besides cats have 4 legs, so we cannot assume that Fido is a cat until we have more information. That is logic: using information to prove a point, but sometimes you realize that you don’t have enough information yet, and the point you prove could be wrong. Another use of logic is in balancing equations. A very simplified example is 7-2=5; if we add 2 to each side, we’ll see 7-2+2=5+2 or 7=7, a true statement. What we do to one side of an equation must also be done to the other side to keep it balanced, as if the equals sign was the pivot point on a balancing scale.
If your student is struggling with any of these building block skills, back up and practice the previous block’s skills until they are mastered. Recall of these facts should come as easily as a reflex action before the student is ready to move on successfully to the next building block. Don’t worry that other students may be moving ahead already—they may not be ready either, and their “progress” will soon result in more struggles. Remember that a student who cannot do division does not know multiplication facts well enough. A student who struggles with multiplication does not know addition facts well enough, and neither does the student who struggles with subtraction. A student who has trouble with addition does not understand place value or number values well enough. Success in math is achieved by mastering skills in sequence and building a solid foundation with each skill before attempting more challenging skills.