Representation Standard


Facility in math is recognized by educators as being key to later success in life. The National Council of Teachers of Mathematics (NCTM) has set ten content standards for the teaching and learning of mathematics from prekindergarten through twelfth grade.

            The standards in this series refer to the entire range of grades. Examples, however, are for prekindergarten to second grade, which includes the grade I teach.

            The NCTM publication Principles and Standards for School Mathematics has complete explanations of these. For more information, you may visit NCTM at

            Bullet points are quotations from the publication. Underneath them are my suggestions for parents.

Math principles and standards, part 10

Representation Standard

            The instructional programs should enable all students to:

  • Create and use representations to organize, record, and communicate mathematical ideas;
  • Select, apply, and translate among mathematical representations to solve problems;
  • Use representations to model and interpret physical, social, and mathematical phenomena.

            When a child holds up two fingers to show how old he is, he is probably showing his first representation of a number with a physical object. Moving into the early school years, children learn to represent numbers in other ways as well: by speaking, writing, gesturing, and drawing. Some use numbers that we can easily recognize because they are standard; others invent their own symbols to get their point across.

            Parents can be most helpful by listening carefully to what their children say about their mathematical observations. In the event that their children have math homework, parents can encourage their children to find a way to put their thoughts onto paper as they figure out math problems. Then, after this is completed, parents ask the kids to use what they wrote to explain their thought process to the adult.

            When family members are trying to figure out a mathematical situation as a group, parents explain to the older siblings the need to respect the budding math talent of the younger ones. There has to be a safe environment in which all can explain the various ways of doing the math. Children at different stages of development will find ways to solve problems that reflect their own level of understanding.

            For example, a third-grader who has learned times tables may easily understand and explain that 6 X 5 = 30. At the same time a first grader may show her understanding by writing 5 + 5 + 5 + 5 + 5 + 5 and then counting, “5, 10, 15, 20, 25, 30” and come up with the same answer, or by adding pairs of the fives into three separate tens, then counting, “10, 20, 30.” Parents would do well to respect that there are many ways - not just one “right” way - to express oneself mathematically and come up with answers to questions in these situations.

            Jay Davidson lives in Palo Alto and has been teaching in San Francisco for 31 years; he teaches first grade. His book, Teach Your Children Well: A Teacher’s  Advice for Parents, is available for $12.95 at Each part of this series is archived and available on his website,


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