How Can We Improve the Teaching of Fraction Concepts?

Cheryl McAllister, SoutheastMissouriStateUniversity

Problem One: There is no standard way to define a fraction. Which of these are fractions?

A. Some of the ways to conceptualize or define fractions include:

  1. Part-whole: means 3 parts out of 4 equal parts of a unit whole
  2. Operator: of something – multiply by 3 and divide by 4 OR divide by 4, then multiply by 3.
  3. Ratio and Rates: means 3 parts compared to 4 parts
  4. Quotient: means 3 divided by 4. is the amount each person receives when 4 people share a 3-unit of something (concrete model)
  5. Measure: means a distance of 3 one-fourth units from 0 on the number line or 3 one-fourth units of a given area

B. The Part-whole concept of fractions is traditionally presented in school mathematics

Problem Two: Students don’t always grasp the whole number concepts that are essential in understanding and performing operations on fractions

1. Greatest Common Factor (Divisor)

2. Least Common Multiple

3. Measurement and Partitive division

4. Operations on whole numbers

Problem Three: Current mathematics curriculums often focus on the procedural aspects (algorithms) for working with fractions, but do little to help student develop a deep, conceptual understanding of fractions.

A. Rational number Project:

Scroll down and click on either the chronological bibliography or the bibliography by author for a wealth of information on teaching fractions. Here are two excellent resources:

Cramer, K., Behr, M., Post T., Lesh, R., ( 1997) Rational Number Project: Fraction Lessons for the Middle Grades - Level 1, Kendall/Hunt Publishing Co., Dubuque Iowa.
Cramer, K., Behr, M., Post T., Lesh, R., ( 1997) Rational Number Project: Fraction Lessons for the Middle Grades - Level 2, Kendall/Hunt Publishing Co., Dubuque Iowa.

Excellent source for a curriculum that builds conceptual understanding prior to teaching algorithms.

Problem Four: Many of the current teachers in our US classrooms don’t understand fractions well enough to teach the topic conceptually

Brown, S. I., Cooney, T. J., & Jones, D. (1990). Mathematics teacher education. In W. R.

Houston (Ed.), Handbook of research on teacher education: a project of the Association of Teacher Educators (pp. 639 – 656). New York: MacMillan Publishing Company.

Carpenter, T. P., Corbitt, M. K., Kepner, H. S. Jr., Lindquist, M. M., & Reys, R. E. (1981).

Results from the second mathematics assessment of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of

fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates, Publishing.

McDiarmid, G. W., Ball, D. L, & Anderson, C. W. (1989). Why staying one chapter ahead

doesn’t really work: Subject-specific pedagogy. In M. Reynolds (Ed.), Knowledge base for beginning teacher (pp. 193 – 205). Oxford: Pergamon Press.

Post, T. R., Harel, G, Behr, M., & Lesh, R. (1988). Intermediate teachers’ knowledge of rational

number concepts. In Fennema et al. (Eds.), Papers from First Wisconsin Symposium for Research on Teaching and Learning Mathematics (pp. 194-219). Madison, WI: WisconsinCenter for Education Research. Retrieved July 21, 2004 from

Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The

case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-21.

Problem Five: How do we help our students develop conceptual understanding of fractions.

1. Takes time

2. Consider this learning model:

Lesh’s Model of Multiple Modes of Mathematical Representation

Fraction Manipulatives Activity

You will work in groups of 3-4 students. Each group will be given a type of manipulative used to represent the fractions. Work through the problems using the manipulatives to represent your thinking and answer the questions below for your particular type of manipulative. Be prepared to present to the entire class how you would use this type of manipulative to teach various fraction concepts. Each group should turn in one copy of the work on all parts with everyone’s name included.

1. On a piece of paper, trace the piece (or pieces) you decide will represent the ‘whole’ or ‘unit’, then trace the piece (or pieces) that represent the following fractions: ½, 1/4, 1/3, 1/6, ¾, 2/3, 5/6.

Example: This would be the work for pattern blocks and 1/6

2. Use your pieces to represent ½ + 1/3. Draw that representation on a piece of paper. Develop a word problem that goes with this math fact.

3. Use your pieces to represent 5/6 – 2/3. Draw that representation on a piece of paper. Develop a word problem that goes with this math fact.

4. Use your pieces to represent 3 x ¾. Draw that representation on a piece of paper. Develop a word problem that goes with this math fact.

5. Use your pieces to represent 2/3 x 3/4 Draw that representation on a piece of paper. Develop a word problem that goes with this math fact.

6. Use your pieces to represent ¾ ÷ 1/2. Draw that presentation on a piece of paper. Develop a word problem that goes with this math fact.