How Can We Improve the Teaching of Fraction Concepts?
Cheryl McAllister, Southeast Missouri State University
cjmcallister@semo.edu
http://cstl-csm.semo.edu/mcallister/mainpage
Problem One: There is no standard way to define a fraction. Which of these are fractions?
2  5 x  3
1.3
, ,
,
, 3  4, .05,
3 5 20 x  2
2.9
A. Some of the ways to conceptualize or define fractions include:
1. Part-whole:
2. Operator:
3
means 3 parts out of 4 equal parts of a unit whole
4
3
of something – multiply by 3 and divide by 4 OR divide by 4, then
4
multiply by 3.
3. Ratio and Rates:
4. Quotient:
3
means 3 parts compared to 4 parts
4
3
3
means 3 divided by 4.
is the amount each person receives when 4
4
4
people share a 3-unit of something (concrete model)
5. Measure:
3
means a distance of 3 one-fourth units from 0 on the number line or
4
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:
http://education.umn.edu/rationalnumberproject/default.html
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:
Wisconsin Center for Education Research. Retrieved July 21, 2004 from
http://education.umn.edu/rationalnumberproject/88_11.html
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), 521.
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
Manipulatives
Spoken symbols
Real world
situations
Pictures and
diagrams
Written symbols
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
whole
part
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.