CONSTRUCTING NUMBER LINES: YOUNG STUDENTS EXPLORE CONCEPTS OF
UNITS, MEASUREMENT, AND PROPORTION M4YC @ Humberwood Downs
Slide 2
Introductions
Slide 3
Our Learning Journey Focus on Number Sense
Difficulties/Challenges Big Concepts Number Knowledge Test
Questions for Clinical Interviews Day 1
Slide 4
Number Knowledge Test A test that is designed to measure the
intuitive knowledge of numbers that an average child has available
at the age levels of 4, 6, 8, and 10 years. Administered by
teachers, graduate students, and researchers from the Dr. Eric
Jackman Institute of Child Study, University of Toronto
Slide 5
Number Knowledge Test
Slide 6
Clinical Interviews
Slide 7
Slide 8
Research Lessons Human Number Line Three students are asked to
line up side by side and the class is asked "Who is in the middle?"
The number of students is increased, so one student is in the
middle, e.g., 5, 7. The students reflect on strategies such as,
having the same number of students on each side determines the
middle. Then a challenge is posed with having 5 students. The
students are asked "Who is in the middle?" At this point the
students reflect on the fact that a person is not the middle, but a
space, with the same reasoning that having the same number of
students on each side determines the middle. In Kindergarten we
used a marker to show the person in the middle, such as a picture
of a sun placed above the person in the middle.
Slide 9
Research Lessons Human Number line
Slide 10
Research Lessons Tim Hortons Lesson The Tim Hortons exploratory
lesson was one of the early lessons to consolidate the students
concept of "middle as halfway". The students enjoy "Timbits," and
this was used as prior knowledge to extend their understanding of
"middle" and to link " middle" and halfway to distance. Butcher
paper was used to represent the road from the school to the zoo. A
stop had to be made halfway so we could buy some Timbits. Students
were to estimate and mark the middle on the road, and then use the
folding strategy to prove or disprove their estimate.
Slide 11
Research Lessons Tim Hortons Lesson
Slide 12
Research Lessons Connecting to the Hundred Chart In an effort
to connect the understanding of a hundred chart to a number line,
we used the hundred chart to find the middle, using the strategy of
the same number of "things" on each side determines the middle. The
challenge was posed when we compared finding the middle with and
without the zero. The students had many reflections including the
"main middle", indicating a group of numbers (4,5,6) are in the
middle of a number line from 0-10, but one was in the "main middle"
(5).
Slide 13
Research Lessons Connecting to the Hundred Chart
Slide 14
Research Lessons The Clothesline The clothesline lesson was one
of the early exploratory lessons to give students practice in
locating "middle", "proportional spacing", and to consolidate their
understanding of "middle as halfway" and to develop spatial
reasoning. A clothesline was made with numbered 3x5 cards and
clothes pegs. Students were told that a strong wind came and blew
all the clothes off the line and only zero and ten remained on the
line. The number five card was replaced on the line, and students
were asked to help put the clothes back on the line. The emphasis
of this lesson was locating middle and equal spacing of the
numbered cards on the clothes line.
Slide 15
Research Lessons The Clothesline
Slide 16
Research Lessons The Freezie Lesson The Freezie lesson was to
provide students practice in locating "middle" to determine "half
of a whole". A freezie was shared for Ms. Hassen and Ms. James. The
students instructed the teacher to cut the freezie in two, so the
teacher purposefully cut it in 1/4 and 3/4. The students were
furious. They said, "No, Ms. James, you made a mistake, you must
give Ms. Hassen a half." The students then suggested ways to find
the middle of the freezie. It was then measured with a ruler to
locate, the "middle" and to find a "half". The students, Ms Hassen,
and myself then had a freezie treat.
Slide 17
Research Lessons The Freezie Lesson
Slide 18
Research Lessons Solving Addition & Subtraction Problems
Children worked with counters and with the number line. The number
line displays counting and measurement simultaneously. Using the
number line gives students a visual image of the operation being
done and a better understanding of the answer. Also, the number
line helps students experiment with decomposing numbers. Benefits:
Over time, this will enable students to become more capable in
performing mental computations. Numbers can be decomposed and the
subunits or smaller amounts can be added or subtracted in varying
orders, yet still be equivalent.
Slide 19
Research Lessons Solving Addition & Subtraction
Problems
Slide 20
Research Lessons Making Rulers
Slide 21
Slide 22
What we learned The linear representation of numbers, which
includes presenting numbers on a number line, can access
foundational mathematical concepts including; ordinality, spatial
awareness and magnitude, concept of middle. Originally focused on
counting with precision. Then spatial awareness was greatly
influenced by their understanding of middle. Finally students paid
more attention to space as well as number counting
Slide 23
What we learned Number line offers insights and supports:
Counting and counting on, counting down Sequencing Understanding
units as equal intervals of distance Use of benchmarks Sense of
proportional reasoning Understanding measurement concepts
Slide 24
Researchers study the development of estimation on number line
Many JK students, even those who can count perfectly from 1 to 10,
do not understand the rank order of the numbers magnitudes. These
childrens number line estimates correlate only minimally with the
magnitudes of the numbers they are estimating.
Slide 25
Researchers study the development of estimation on number line
Even after children learn the rank order of numbers magnitudes,
they still do not immediately represent the magnitudes as
increasing linearly. Their number line estimates often increase
logarithmically with the size of the number being estimated. By
grade 2 children with experience their magnitude estimates increase
linearly
Slide 26
Todays Lesson Now its your turn to create your VERY OWN ruler!
You will have the opportunity to make three rulers, all will begin
with 0. You can choose how long your ruler is going to be AND even
more exciting youre going to be able to choose what number you want
as your end point on ONE of your rulers. The other two will be 0
10. The only rule is that you have to use a marker to write the
zero and the end number. Everything in between you can use a
pencil. Lets Try!
Slide 27
Questions for Observers 1. What specific strategies did you
notice as students were building their rulers? 2. Do students focus
on equal spacing? What strategies do they use to determine the
intervals? 3. When making their own rulers, were students able to
use an ordinal counting strategy in tandem with their spacing
strategy? 4. Did you notice any students using the middle as a
benchmark to create the spaces? If so, how did they find middle?
How did children show their understanding of half? 5. Did you see
evidence of students using gestures and actions to express their
mathematical thinking to a peer, the group, etc.? Please describe
what you saw. 6. Is there a difference between boys and girls with
their accuracy in spacing numbers on the number line/ruler? 7. Do
children change the orientation of their rulers? 8. Did you see
children imitating their peers? Did the imitating strategy seem to
help students?