To learn more about TIAN, or the Adult Numeracy Center's other professional development, curriculum development, or consulting services, contact:
(617) 873-9600

or visit our website:







TIAN Bundle 2 Articles and References (For Teachers) about Strengthening Operation and Symbol Sense


About Operation Sense


  • Gregg, J. & Gregg D. (May 2007). Measurement and Fair-Sharing Models for Dividing Fractions. Mathematics Teaching in the Middle School, 12(9), 490-496.
    Includes examples of how to move students from division of whole numbers to division of fractions.
  • de Silva Lamberg, T. (September 2007). Student Approaches to Unitizing in Fair-Share Problems. Mathematics Teaching in the Middle School, 13(2),114-116.
    This short article analyzes four students' methods for dividing (fair-share situation).
  • Nugent, P. (September 2007). Lattice Multiplication in a Preservice Classroom. Mathematics Teaching in the Middle School, 13(2),110-113.
    Lattice multiplication is an alternative algorithm for teaching multiplication that helps students see the place values.
  • Taber, S. (December 2006). Using Alice in Wonderland to Teach Multiplication of Fractions. Mathematics Teaching in the Middle School, 12(5), 244-249.
    Although practitioners won't be reading Alice in Wonderland to her students, this article does provide examples of the difference between proportional and nonproportional thinking with fractions.
  • Tent, M. (August 2006). Understanding the Properties of Arithmetic: A Prerequisite of Arithmetic. Mathematics Teaching in the Middle School, 12(1), 22-25.
    Provides strategies for teaching the commutative, associative, distributive, and identity properties of addition and multiplication.

About the Understanding of the Equal Sign


  • Falkner, K., Levi,L., & Carpenter, T. (1999). Children's Understanding of Equality: A Foundation for Algebra
    This brief article describes the misconceptions students develop at a very early age about the equal sign. Included are strategies one teacher used to try to overcome those misconceptions.
  • Freiman, V. & Lee, L. (2004). "Tracking Primary Students' Understanding of the Equality Sign". Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (vol.2, pp. 415-422). 
    This research was aimed at tracking young students' understanding of the equal sign by monitoring how they responded to simple problem types such as a + b = c and c = a + b. Types of errors in children in K, grade 3, and grade 6 were then analyzed.
  • Knuth, E. & Stephens, A. (July 2006). Does Understanding the Equal Sign Matter? Evidence from Solving Equations. Research in Mathematics Education, 37(4), 297-312.
    This research project investigated the relationship between being able to solve simple algebraic equations and an understanding of the equal sign.

Other Related Articles


  • Algebraic Thinking from Assessment Resource Banks: English, Mathematics, and Science.
    This collection of mini-resources includes hands-on strategies for teaching equality and the additive, commutative, and associative properties, beginning very early on in learners' math education.
  • Computation, Calculators, and Common Sense (May 2005). 
    A position paper from the National Council of Teachers of Mathematics in reaction to the question of calculator use in the classroom.
  • National Research Council (2001). "Number: What Is There to Know?" from Adding It Up: Helping Children Learn (71-114).
  • Schifter, D., Bastable, V., Russell, S.J., Riddle, M. & Seyferth, L. (2008). Algebra in the K-5 Classroom: Learning Opportunities for Students and Teachers. In Greenes, C. E. and Rubenstein, R. (Eds.), Algebra and Algebraic Thinking in School Mathematics: 70th Yearbook (pp. 263-277). Reston, VA: NCTM.
    Discussion of classroom investigations into how students develop algebraic thinking.
  • Stiff, L. (April 2001). Making Calculator Use Add Up from NCTM News Bulletin 37, Vol. 3
    A response to a newspaper article about the use of calculators in the classroom.