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发帖数: 19398 | 1 幼教数学 What is Early Math and Why Should We Care?
[Posted by Ben Braun]
By Jennifer S. McCray, Assistant Research Scientist and Director of the
Early Math Collaborative at Erikson Institute [SEE http://www.erikson.edu/early-math-collaborative/ -- See this site]
Effective early childhood math teaching is much more challenging than most
people anticipate. Because the math is foundational, many people assume it
takes little understanding to teach it, and unfortunately this is distinctly
not the case. In fact, the most foundational math ideas - about what
quantity is, about how hierarchical inclusion makes our number system work,
about the things that all different shapes and sizes of triangles have in
common - are highly abstract ones. While we should not expect or encourage
young children to formally recite these ideas, they are perfectly capable of
grappling with them. Further, they need to do so to develop the kind of
robust understanding that will not crumble under the necessary memorization
of number words and symbols that is to come in kindergarten. In preschool,
before there is really any opportunity for "procedural" math, it is
important that we give children ample opportunity to think about math
conceptually. In this essay I will discuss several profound ideas from
early childhood mathematics, including examples of effective early math
classrooms. Along the way I will share some of the resources that my
colleagues and I have developed to help early childhood educators develop as
skillful teachers of early mathematics.
About Early Math
As a doctoral student, I first got interested in early mathematics by way of
cognitive science. I fell in love with the precise and thoughtful
cognitive developmental work that built on what Jean Piaget had begun.
Through clever experimental designs and a careful parsing of concepts over
the last 40 years, developmental psychologists have made enormous strides in
understanding how the mind develops during childhood. Many of their
findings have profound implications for mathematics, and since my degree was
to be in applied child development, early math education provided a way to
make studying cognitive development useful to me.
As it turned out, early math was a useful place to put energy for far more
important reasons. In a now-landmark study in 2007 [1] using six
longitudinal data sets, Duncan et al. found that math concept understanding
at kindergarten entry predicted not only later math achievement, but also
later reading achievement; reading at kindergarten entry, however, did not
predict later math. This finding was replicated in a large-scale Canadian
study in 2010 [2], which found that early math skills were stronger
predictors of general academic success than either reading skills or social-
emotional skills at school entry. We don't yet know for certain why this
association is so strong, but it is at least clear that early math is
important. It is also true that the differences we observe in math
achievement at kindergarten entry tend to fall along socio-economic lines,
so alleviating those differences relates to issues of educational equity.
Early math was also a useful focus because of the pronounced need (in the U.
S. especially) for improved instruction in mathematics in preschool and
early elementary settings. Years after the seminal work by Deborah Ball [4]
on the need for improved pedagogical content knowledge, and by Liping Ma [3]
on the lack of a "profound understanding of fundamental mathematics" among
later-grade elementary teachers, math educators turned their lens to those
teaching our youngest students. It turns out that students of teacher
education who "love kids but hate math" are commonly directed by faculty to
teach in the younger grades. This has left us with a preponderance of
preschool and primary teachers who are both underconfident and underprepared
in mathematics teaching.
Teaching Early Math
So what does mathematics teaching look like in a preschool classroom?
Recall first that preschool means children between the ages of 3 and 5, and
that their range of normative development is exceedingly wide. In this
group of kids there will be children who are not "potty-trained" alongside
children who have begun to read, so teachers have to cast a very wide net.
Further, for this age group, "teaching" is something that is often done only
when all the heavy lifting of being sure everyone is comfortable, rested,
and not in tears is complete. Sit-and-listen techniques are effective only
when the content is exceedingly entertaining - as in a story is being read -
and the children have very limited capacity for absorbing information
directly from text, and less-limited but still primitive abilities to
communicate their own ideas.
For these reasons, learning in early childhood classrooms consists almost
entirely of "active learning." In fact, early childhood has a long and proud
connection to the type of teaching that emphasizes student-directed/teacher
-facilitated activities. Child choices and the use of prepared "centers"
are favored, with limited time spent on whole group activities of any kind (
"circle time" being the exception), and small groups being occasionally led
by a teacher. This is not an environment that is amenable to worksheets,
and for that, early childhood teachers are generally extremely grateful. It
also means, however, that whatever content is introduced comes fairly
directly from the intentions and understandings of the teacher, who designs
and facilitates experiences that lead children to construct new thinking.
Some Useful Interventions
Given this learning environment, my colleagues and I decided to focus our
work on improving teachers' understanding of the early math content they
should be working into their interactions with young children. By studying
the cognitive developmental and early math education literatures, we
developed 26 Big Ideas [SEE http://earlymath.erikson.edu/about-early-math-programming-for-teachers-and-teacher-educators/approach/ ] that we wanted to be sure early childhood teachers understood well and knew how to address. One example is the idea that "any collection of objects can always be sorted in more than one way." While this is not a conventional mathematical idea, it is foundational to the types of thinking that underlie our experience of sets (there are 6 pieces of fruit; there are 2 apples, 2 lemons, and 2 bananas; there are 2 red pieces of fruit and 4 yellow pieces of fruit) and therefore an important understanding for young children to see, explore, and experience. It has generative implications for understanding number and algebra in later life, and helps children flex and develop their logical thinking skills.
To help teachers make such an idea come to life, we developed what we call "
Research Lessons." These are skeleton lesson plans for activities teachers
can use over the period of a month or more (through slightly altered
iterations). For the Big Idea above, we ask teachers to conduct a read-
aloud of a beautifully illustrated children's book called Five Creatures by
Emily Jenkins. In the book, a family of two adults, one child, and two cats
is described differently from page to page, as in "In my family, there are
five creatures?three who like milk, one who does not, and one who only
drinks it in coffee?three with orange hair (child, one adult, one cat), one
with gray hair, and one with stripes?" This book is read several times over
a period of days, with lots of discussion. At some point, the teacher
introduces two large circles, drawn out on the rug with tape: half the
class are the "creatures" and half are the audience. Together, teacher and
audience sort the "creatures" using binary (A/B) sorting to place them
inside the circles, as in "the creatures with long hair and the creatures
with short hair" or "the creatures with white in their shirts and the
creatures without white in their shirts." This leads to useful discussions
about shared definitions for categories and sometimes generates the (
exciting!) need for a third circle.
Conclusion
While it often goes unrecognized, the need for strong early math skills
among children and early childhood educators is strong. Early math is
highly abstract, and is a key indicator of later school success. What
happens in preschool and early elementary classrooms has a direct impact on
students for the rest of their educational experiences, from elementary
school through postsecondary work. Our early childhood teachers need better
preparation and in-service training to understand their crucial role in
mathematics education. We will best be able to rise to the challenges of
early math education through collaborative efforts involving teachers,
teacher educators, and mathematical scientists.
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NOTE: YOU CAN SEE THE RESEARCH PAGES REGARDING BEST PRACTICES AT http://earlymath.erikson.edu/research-prek-and-elementary-school-research-projects/
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References
[1] Duncan GJ, Dowsett CJ, Claessens A, Magnuson K, Huston AC, Klebanov P,
Pagani LS, Feinstein L, Engel M, Brooks-Gunn J, Sexton H, Duckworth K, Japel
C. "School readiness and later achievement." Dev Psychol. 2007 Nov;43(6):
1428-46.
[2] Pagani, Linda S. et al. "School Readiness and Later Achievement: A
French Canadian Replication and Extension." Developmental Psychology. Vol.
46(5): 984-994. September 2010.
[3] Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers'
Understanding of Fundamental Mathematics in China and the United States.
Routledge, 1999.
[4] Ball, D.L. (1988). Knowledge and reasoning in mathematical pedagogy:
Examining what prospective teachers bring to teacher education. Unpublished
doctoral dissertation, Michigan State University, Lansing.
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发帖数: 19398 | 2 看到中间 "26 big ideas"的时候，我才意识到可能是熟悉的人
再回过头一看，果然。11月中刚刚给作者做过围棋数学的 workshop
浮在 ananpig (●○ 围棋数学一把抓的安安猪) 的大作中提到: 】
distinctly
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