This free Information Age Education Newsletter is written by Dave
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This is the 17th of a
series of IAE Newsletters exploring educational aspects of the current
cognitive neuroscience and technological revolution. Bob Sylwester
(Newsletter # 75) and Dave Moursund (Newsletter # 76) provided two
introductory newsletters. Newsletter # 77 and subsequent newsletters
were written by guests. However, Sylwester and Moursund also
contributed to this emerging collection.
For the most part, the guest newsletters focus on cognitive
neuroscience. Dave Moursund provides Information and Communication
Technology follow-up commentary to the articles. In addition, readers
are invited to send their comments using the Reader Comments directions
near the end of each newsletter.
We encourage you to tell your colleagues and students about the free
IAE Newsletters. Free back issues and subscription information are
available at http://i-a-e.org/iae-newsletter.html.
Cognitive Neuroscience, Computers,
and Math Education
David Moursund
Emeritus Professor, University of Oregon
Using brain/mind science and computers to improve elementary school math education is a free book Moursund (2004, 2012). This book has recently undergone careful editing and I added a new chapter.
I found it interesting to analyze the content written eight years ago
from the point of view of what we knew about cognitive neuroscience,
computers, and math education in 2004 versus what we know today. I also
found it interesting to see how math education research and cognitive
neuroscience have helped (and, unfortunately in many cases, failed to
help) improve our math education system during the past eight years.
A Little History
The Sumerians developed reading and writing about 5,200 years
ago. The written language that was developed included symbols for
numbers. Schools were developed to teach reading, writing, and
sometimes a little arithmetic. Reading, writing, and arithmetic have
been part of a literate person’s informal and formal education for more
than 5,000 years.
It is clear that human genetics and early childhood development
predispose us to learn oral communication. Oral fluency provides a
strong foundation for learning reading and writing. While many people
find it difficult to develop a contemporary level of competency in
reading and writing, our educational system is geared to addressing
such difficulties.
In recent years we have come to understand that an infant’s brain has
some innate math-related knowledge and skills and is predisposed to
learn some math-related topics. In his 2000 book The Math Gene,
Keith Devlin argues that the ability to learn a natural language is
closely linked with being able to learn arithmetic. There is no innate
reason why so many students who learn to read and write should end up
hating math and claiming that they cannot do math.
A natural language such as English changes over times. New words are
added, some words fall into disuse, and definitions and usage change.
The amount of literature written in natural languages grows over time,
and some of it lasts for many centuries. And, of course, there is a
steady increase in the totality of written accumulated human knowledge.
Thus, teachers of reading and writing face a continuing challenge of
preparing their students to achieve an appropriate contemporary level
of literacy for their adult lives.
In some sense, the discipline of math grows more rapidly than the
reading and writing domains of the language arts. Math is a
vertically structured discipline in which the creation of new math
knowledge and skills is built on thousand of years of accumulated math
research. The steadily growing use of math in the sciences, economics,
business, and many other disciplines creates a math education challenge
that is quite different from the types of challenges faced in Language
Arts education. While relatively few people find the need to solve a
quadratic equation, graph a polynomial function, calculate the
correlation between two sets of data and follow an argument based on
statistical analysis, or prove a geometric theorem in their everyday
lives, our educational system has decided that all students need to
study such topics in order to graduate from high school (CCSS, 2012).
We want today’s students to learn topics from algebra, geometry,
probability, and statistics—subjects that had not yet been discovered
back when the first schools were created. Electronic calculators and
computers represent still newer content and powerful new aids to doing
math, and these can be integrated into a math curriculum. Thus, math
curriculum specialists are faced by a continually changing challenge of
what to include in the math curriculum, what math all students should
study, and what math is needed for various careers and for further
study of math.
We have had thousands of years of experience in helping children learn
reading, writing, and simple arithmetic. However, we have had only
modest experience in trying to meet requirements that all students
should study algebra in the eighth grade and learn various topics from
geometry, probability, and statistics before they complete high school.
Currently, now more than 50 years into the Information Age, we still
have not yet decided on what calculator and computer content to
thoroughly integrate into the K-12 math curriculum or how to assess
student learning of this new aspect of a math curriculum.
As infants are learning their native language(s), their
parents and other caregivers often speak in “motherese” and keep the
vocabulary quite simple. However, a young child is also immersed in an
environment of adult conversations that include vocabulary, ideas, and
experiences far above his or her current language development levels. A
child’s language development is pushed by being in such a “rich”
language environment.
This same thing happens in math, but there is a major difference.
Although math is a language, not much math is spoken in everyday
conversation, and the math that is spoken to young children is often
not yet relevant to a child’s life.
I grew up in a household in which both my mother and father had
advanced degrees in math and taught math at the college level. My young
brain was routinely exposed to math content conversations and math
thinking. I entered kindergarten having grown up in both a rich natural
language environment and a rich math language environment. This early
head start has served me well throughout my life.
Piaget and other researchers developed the field of cognitive
development (McLeod, 2009). Quite a bit of Piaget’s work has stood the
test of time and/or served as a good starting point for more modern
research. The rate of cognitive development varies among students. The
rate of development depends on a combination of nature and nurture.
Moreover, cognitive development in math does not necessarily progress
as rapidly as does overall cognitive development.
Piaget’s four basic stages of cognitive development are sensorimotor
(birth to age 2), preoperational (ages 2 to 7), concrete operations
(ages 7 to 11), and formal operations (ages 11 and beyond). At the
formal operations level, children begin to develop a brain/mind that
can deal with the type of abstractions that are fundamental to
mathematics. However, even in kindergarten, students are being exposed
to some of the abstract notation, vocabulary, and nuances of math. For
math students who have grown up in a math ”poor” environment, the math
that is being presented is considerably above their level of math
cognitive development.
Like any curriculum, math has both breadth and depth. In some sense, a
new “breadth” topic is a leveler. Many students studying the new topic
are essentially starting from scratch, and the teacher does not assume
a great depth of prerequisites. However, when a topic is designed to
add depth to a student’s math knowledge and skills, the teacher and
curriculum make assumptions about the prerequisite math knowledge,
skills, and math cognitive development of the students. The students
who don’t meet the prerequisites are apt to be in way over their heads.
This frequently leads to a rote-memory learning approach, with little
underlying understanding on the part of the student. That, in turn,
leads to the student falling further behind when a new “depth” topic is
taught that assumes an understanding of previous topics.
The Past Eight Years
Our math education system has made a number of changes since I was a
child. Still, to me it seems that the system exhibits considerable
resistance to change. For example, in 1979 the National Council of
Supervisors of Mathematics and in 1980 the National Council of Teachers
of Mathematics strongly supported the integration of calculators into
the elementary school math curriculum.
In those days, calculators were still rather expensive and somewhat
fragile. Now, more than 30 years later, calculators are very
inexpensive, use solar-powered batteries, are quite rugged, and are
routinely used by adults. However, many elementary school teachers
still strongly resist their use in school. Where calculators are
allowed on state and national tests, the test questions are usually
designed so that a student gains very little advantage in using a
calculator. The newly developed Common Core State Standards (CCSS) in
math place increased emphasis on understanding, more emphasis on depth
in a less broad curriculum, and little emphasis on use of calculators
and computers as an aid to problem solving (CCSS, 2012). In contrast,
the CCSS standards being created for science have drawn considerable
criticism because they place very little emphasis on use of computers
in science.
During the past eight years many schools have explored the idea of
having classroom sets of computers and/or computer tablets. Some
schools and school districts have acquired one laptop or tablet
computer per student, and many allow students to carry them home.
However, the big push for laptop and tablet computers in our K-12
schools is mainly for their use in computer-assisted learning, distance
learning, and information retrieval. There has been only very modest
progress in the integration of these powerful Internet-connected tools
as aids to representing and solving math problems. Little progress has
occurred toward allowing laptop and tablet computers on state and
national math tests.
The past eight years have brought us considerable advances in understanding the learning disability dyslexia (a major challenge to learning to read) and the learning disability dyscalculia
(a major challenge to learning arithmetic). There is a high level of
co-morbidity between dyslexia and dyscalculia (Butterworth, 2005).
Our schools have made good progress in early detection of dyslexia and
other reading problems. Early and strong interventions often occur. The
same cannot be said for the math learning difficulties that students
encounter because of some combination of dyslexia, dyscalculia, and
other math-related learning disabilities. This is in spite of the fact
that we have made good progress in understanding some of the brain
functions specifics of dyscalculia.
Here is one of my favorite quotes:
“When you spoke of a
nature gifted or not gifted in any respect, did you mean to say that
one man may acquire a thing easily, another with difficulty; a little
learning will lead the one to discover a great deal; whereas the other,
after much study and application no sooner learns then he forgets …”
(Plato, 428/427 BC– 348/347 BC.)
There is considerable research literature on forgetting and ways to teach and learn that will decrease forgetting. (See http://frank.itlab.us/forgetting/.)
While CCSS (2012) emphasizes learning for understanding, our steadily
increasing emphasis on high stakes testing is causing an increased
emphasis on math rote memory learning that is soon forgotten.
The problem of teaching over the heads of many students—because their
level of math cognitive development and level of math maturity is below
what is needed—has gotten worse. This is being caused by a strong
movement to make algebra a required eighth grade course and the
requirement that students take an increasing amount of math for high
school graduation. To me it seems like the people who are pushing
algebra into the eighth grade and increasing the math requirements for
high school graduation are ignoring what we are learning about math
cognitive development. The work of the van Hieles done more than fifty
years ago showed that even then we understood the problem of putting
students into a math course that was too much above their current level
of math cognitive development (van Hiele Model, n.d.).
For many years preservice teachers have learned about the idea of
students learning reading and writing across the curriculum. We want
students to learn to read well enough in each school discipline so that
they can use their reading skills to further their learning in each
discipline they study in school. During the past eight years I haven’t
seen any progress in having students learning to read math well enough
to make use of reading math as a major aid to learning math. Students
are not learning to make effective use of the math-oriented Web
resources.
David Moursund earned his doctorate in mathematics from the
University of Wisconsin-Madison. He taught in the Mathematics
Department and Computing Center at Michigan State University for four
years before joining the faculty at the University of Oregon.
At the University of Oregon he taught in the Mathematics Department,
served six years as the first Head of the Computer Science Department,
and taught in the College of Education for more than 20 years.
A few highlights of his professional career include founding the
International Society for Technology in Education (ISTE), serving as
ISTE’s executive officer for 19 years, and establishing ISTE’s flagship
publication, Learning and Leading with Technology.
He was a major professor or co-major professor for 82 doctoral
students. He has authored or coauthored of more than 60 academic books
and hundreds of articles. Many of these books are available free
online. See http://iae-pedia.org/David_Moursund_Legacy_Fund.
He has presented hundreds of professional talks and workshops. In 2007
he founded Information Age Education (IAE), a non-profit company
dedicated to improving teaching and learning by people of all ages
throughout the world. See http://iae-pedia.org/Main_Page#IAE_in_a_Nutshell.
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