MATH
The early years
Developmental Progression Instructional Tasks
was impressive!” Such thinking is one reason
why math is a core component of cognition.
Counter (Small Numbers) Cubes in the Box: A child counts a small set of cubes.
Accurately counts objects in a line Put them in the box and close the lid. Then ask the
What we know
to 5 and answers the “how many” child how many cubes you are hiding. Tip them out
●
question with the last number and count together to check.
Learning math at an early age is
counted. When objects are visible, Road Race Counting Game: Students identify number
critically important for young children,
and especially with small numbers, amounts (from one to fi ve) on a dot frame and move
especially those from disadvantaged
begins to understand cardinality. forward a corresponding number of spaces on a
communities.
●
“1, 2, 3, 4… four!” game board.
Educators often underestimate what
young children know and can learn
Counter and Producer (10+) Counting Tower: Allow children to count to 20 and about mathematics.
Counts and counts out objects beyond. Ask them to make towers with objects such ● Using research-based learning
accurately to 10, then beyond. as coins. Children should build a tower as high as they trajectories is effective in promoting
Keeps track of objects that have can, placing more coins, but not straightening coins math learning.
and have not been counted. already in the tower. The goal is to estimate and then
Counts a scattered group of 19 count to fi nd out how many coins are in your tallest
chips, keeping track by moving tower. To count higher, have children make pattern
About the authors
each one as they are counted. “walls.” They build a pattern block wall as long as they
Douglas H Clements is SUNY Distinguished
can. This allows them to count to higher numbers.
Professor and Julie Sarama Associate
Professor of Learning and Instruction at
Figure 1: Examples of Selected Levels from the Building Blocks Learning Trajectory – see more
the University at Buffalo, State University
examples at www.betterevidence.org
of New York. They conduct research on
young children’s learning of math, geometry
years of elementary school engage in math
●
Instructional tasks: The paths of teaching. education, and the scale-up of scientifi cally-
far less than they do in literacy. Furthermore, The fi nal part consists of a set of tasks, based curricula and intervention models.
too many curricula and programs for young matched to each of the levels of thinking
children “teach” too much of what they already in the developmental progression. These Author note
know. There are examples of good practice, tasks are designed to help children learn This paper was based upon work supported
but we can and must do better. High-quality the ideas and skills needed to achieve that in part by the Institute of Educational
early education results in learning benefi ts level of thinking. That is, teachers can use Sciences (U.S. Department of Education,
throughout elementary school, especially for these tasks to promote students’ growth under the Interagency Educational Research
children from disadvantaged communities. from one level to the next. The second Initiative, or IERI, a collaboration of the
column in Figure 1 provides example tasks. IES, NSF, and NICHHD) under Grant No.
Learning trajectories: The secret of R305K05157, “Scaling Up TRIAD: Teaching
success Benefi ts of learning trajectories Early Mathematics for Understanding with
Educators generally agree that teachers Thus, learning trajectories describe the Trajectories and Technologies.”
should “start where the child is” and goals of learning, the thinking and learning
“differentiate teaching.” But how? processes of children at various levels, and
Research has provided a powerful tool: the learning activities in which they might
Further reading
learning trajectories. Students follow engage. Several “gold standard” randomized Clements D H & Sarama J (2009). Learning
natural developmental paths in learning control trial studies have shown that curricula and Teaching Early Math: The Learning
mathematics. When teachers understand and professional development based on Trajectories Approach. New York:
these, and sequence activities based on them, learning trajectories increase children’s Routledge.
they build learning environments that are achievement more than those that do not.
developmentally appropriate and effective.
NCTM (2006). Curriculum Focal Points
Learning trajectories have three parts:
For Prekindergarten Through Grade 8
Educators generally
●
Goals: The big ideas. Goals should include
Mathematics: A Quest for Coherence.
the big ideas of mathematics – clusters of
agree that teachers should
Reston, VA: National Council of Teachers of
concepts and skills that are mathematically ‘start where the child is’
Mathematics.
central and coherent, consistent with
and ‘differentiate teaching.’
National Research Council (2009).
children’s thinking, and generative of future
learning. For example, counting and how to
But how?
Mathematics in Early Childhood:
Learning Paths Toward Excellence and
solve problems using counting. Equity. Committee on Early Childhood
●
Development progressions: The paths of A teacher participating in one of these Mathematics, Christopher T Cross, Taniesha
learning. The developmental progression studies observed one student had almost A. Woods, Heidi Schweingruber, Editors.
is a typical path children follow to achieve fi lled her pretend pizzas with toppings in the Center for Education, Division of Behavioral
their goal. Our learning trajectories provide task she was working on. As she got ready and Social Sciences and Education.
simple labels and examples for each level to roll the number cube, she said, “I’m going Washington, DC: National Academy Press.
of each developmental progression, and to get a high number and win!” “You can’t,”
this is shown in Figure 1. The fi rst column replied her friend, “You have four spaces
Sarama J & Clements D H (2009), Early
describes two main levels of thinking in the and the number cube only has ones, twos,
Childhood Mathematics Education
counting learning trajectory (there are many and threes on it.” The teacher reported, “The
Research: Learning Trajectories for Young
more before, in between, and after). numbers may be small, but the reasoning
Children. New York: Routledge.
fall 2009 Better: Evidence-based Education 7
Better(US) Fall09.indb 7 14/10/09 13:04:48
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