METHODS

1

The Developmental Progression of Rational Number
Understanding Using the Number Line
Stephanie Hanson, Diana Chang, Michelle Chee,
Niki Dowlat Singh, Allison Musson, and Joan
Moss Dr. Eric Jackman Institute of Child Study,
Ontario Institute for Studies in Education,
University of Toronto
RESULTS/RECOMMENDATIONS
METHODS
PURPOSE

• JUNIOR KINDERGARTEN
• Participants
• 4 JK students from JICS (1 male, 3 females)
• Materials
• A strip of Bristol board for the number line
• Cut-outs of houses labeled with whole numbers 1
  to 5
• Protocol
• Place house number 2, when house numbers 1, 3,
  and 5 are present
• Show where I would end up if I were driving from
  house number 1 to house number 2 and only got
  halfway there

• To explore how childrens understanding of
  rational numbers develops over the elementary
  grades (using the number line)
• Fractions are typically taught using the pie
  strategy. This limits students abilities to
  comprehend how fractions compare to one another
  and relate to whole numbers
• We were interested in exploring how manipulating
  fractions on the number line can improve
  students conceptual understanding

• EARLY YEARS
• Results
• Students had varying knowledge of half, and only
  one was able to demonstrate one quarter and three
  quarters
• Students paid some attention to even spacing, but
  in relation to the house directly next to the one
  they were placing
• None of the students could identify any numbers
  between 0 and 1
• Three students were able to correctly place the
  house on the halfway point between 0 and 1, but
  they could not identify what the house would be
  called
• Recommendations
• Introduce students to rational number by
  integrating these ideas into every-day activities
• Explicitly teach students from a younger age that
  there are many numbers between 0 and 1 and
  demonstrate this on a number line
• Expose students to the written representation of
  fractions earlier so that they can develop a
  conceptual understanding of it

SUPPORTING LITERATURE

• GRADE ONE
• Participants
• 4 Grade One students from JICS (2 males, 2
  females)
• Materials
• A strip of Bristol board for the number line from
  0 to 1
• Cut-outs of houses (unlabeled)
• Protocol
• What if someone wants to build a new house
  halfway between house 0 and house 1, where would
  the house be built?
• If a car is driving from house 0 to house 1 and
  they break down here (at ¼ and then ¾), where did
  they end up?

• Placing fractions on a number line is crucial to
  students understanding because it allows
  students to
• further develop their understanding of fraction
  size
• see that the interval between two fractions can
  be further partitioned
• see that the same point on the number line
  represents an infinite number of equivalent
  fractions
• (Lewis, p43)
• In the Japanese Curriculum, the number line model
  is used to help students recognize fractions as
  numbers and learn how they are related to whole
  numbers
• (Watanabe, 2007)

• GRADE THREE
• Participants
• 4 Grade Three students from JICS (2 males, 2
  females)
• Materials
• Two strips of Bristol board for the number lines
  labeled 0 to 1 (one calibrated by tenths, and one
  uncalibrated)
• Pointers labeled with fractions
• Protocol
• Can you show me where ½ is on the number line?
  Can you locate ¾ on the number line? What about
  7/8? Do you think that it is between ½ and ¾ or ¾
  and 1? Why?

• PRIMARY/JUNIOR
• Results
• Students were generally able to identify and
  compare benchmark fractions on the number line
• Use of partitioning (concretely with hands)
  during explanations by some students
• Generally limited understanding of equivalency
• Recommendations
• Build conceptual understanding of part-whole
  relationships (e.g., using fraction strips) and
  how they are related to the written
  representation of fractions
• Help students develop an understanding that two
  equivalent fractions can occupy the same point on
  the number line through layering and fraction
  wall activities

• GRADE FIVE/SIX
• Participants
• 4 Grade Five/Six students from JICS (2 males, 2
  females)
• Materials
• Two strips of Bristol board for the number lines
  labeled 0 to 1 (one calibrated by tenths, and one
  uncalibrated)
• Pointers labeled with fractions
• Protocol
• Where is 2/6ths located on the number line? Place
  a pointer there. Now can you find 4/12ths. Can
  you think of another fraction that is the same is
  2/6 and 4/12?