Dr. Cotter on Teaching Math

I’d like to take some time to discuss the science and art of teaching math. I call it a science because much research has been done into how children learn in general and especially math. I call it an art because each child is different, requiring the instructor to tweak the lesson to help each individual child.

The teacher’s role

Research has shown that 40% of what a student learns depends upon the teacher. Let’s start with the teacher’s beliefs and attitudes. If the teacher shows math anxiety, the child will probably absorb some of that fear and dread. If the teacher views math simply as a bunch of facts and rules to be memorized, the child’s math education will be built on a shaky foundation. If the teacher believes the myth of “a math person,” the child may decide they do not qualify and stop trying. 

On the other hand: 

  • Fortunate is the child whose teacher is convinced of the importance of math for daily living, future careers, and understanding of our world.
  • Fortunate is the child whose teacher views math problems like solving a puzzle, trying different methods, and looking for several solutions.
  • Fortunate is the child whose teacher realizes there is more than one way to do calculations, some are more efficient than others, and not everything needs to be written down.
  • Fortunate is the child whose teacher abstains from flash cards and timed tests, instead approaches facts using number sense and games.
  • Fortunate is the child whose teacher knows mastery is achieved through thinking, not blindly following an example or practicing some rule over and over.
  • Fortunate is the child whose teacher uses math to help their student develop self-confidence and independent thinking.
  • Fortunate is the child whose teacher understands that some frustration is a normal part of learning and encourages the child to persist.
  • Fortunate is the child whose teacher does not constantly dispense rewards, verbal or otherwise, causing the child to rely on the teacher for assurance, instead of their own thinking, for every step of the way.
  • Fortunate is the child whose teacher is aware that a child develops concentration by being allowed to concentrate and being protected from unnecessary interruptions.
  • Fortunate is the child whose teacher radiates joy and helps the child develop a love of mathematics.

Independent learning 

While it is true that one goal of education is the ability to learn independently, the basics of any discipline must be learned first. Skiing down an expert run before mastering control of the skis would lead to disaster. Competing in a triathlon before learning to swim would lead to a waterlogged failure. Giving a young child a math workbook and expecting them to master elementary math is wishful thinking. They are merely learning how to fill in the blanks. Having a child watch a math video alone is not much better. There is no one to answer questions and no way to assess understanding. There is no one to listen to and encourage the child’s alternate way of thinking. There is no one to cultivate a desire for deep understanding.


If math is viewed as numerous rules and procedures to be memorized, lots of homework might make sense. Anything memorized without understanding needs frequent review to remain in memory. Homework should never be so difficult that the lesson needs to be retaught, or worse yet, the answers are spoon-fed to the child. With RightStart Mathematics, homework is usually playing a game.

The world of math

It is so easy to think that the math we learned as a child is sufficient for our children and grandchildren. The fact is today’s children need different math. They need to learn topics in geometry, equations, probability, statistics, fractals, and combinatorics to name a few. At the same time, arithmetic is less important because of calculators and computers. 

It is also easy to think that the way we learned math, often by rote and without much comprehension was adequate. On the contrary, we now know that a deep understanding of concepts removes anxiety, lessens the burden of memorizing, makes advanced math easier to grasp, and makes math more enjoyable.

Sometimes math is thought of as exclusively as a paper and pencil activity. On the contrary, what we write on paper is a shortcut for expressing a concept, often found in some form in the real world. 

Some of the reasons we read to our children are to foster a love of reading, to enlarge their worlds, and to expand their vocabularies. Similar reasons apply to the realm of math. Use precise vocabulary especially to the younger child: when appropriate, say ellipse, not oval and rhombus, not diamond. Refrain from inferring that squares aren’t rectangles (they are), that you can’t take 7 from 5 (you can: the answer is –2), that the product after multiplying is always greater than either factor (not so: 2 × 0 = 0, which is not greater than 2).

One interesting fact recently discovered by researchers is that when mathematicians discover beauty in math, their brains light up in the same regions as that of artists when they find beauty in art. I hope you and your children have much success in learning and discovering mathematics.



  1. Great article! I am fortunate to love math and am passing that love onto my son. Even so, this article is incredibly encouraging and I’m sending it to other homeschool friends. I especially appreciated your perspective on kids needing to learn different aspects of math today because of computers etc.

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