Information Age Education
   Issue Number 91
June, 2012   

This free Information Age Education Newsletter is written by Dave Moursund and Bob Sylwester, and produced by Ken Loge. The newsletter is one component of the Information Age Education project. See http://iae-pedia.org/ and the end of this newsletter. All back issues of this newsletter are available free online at http://i-a-e.org/iae-newsletter.html.

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.

Conrad Wolfram’ excellent 17-minute TED talk on this topic is available at http://www.ted.com/talks/lang/en/conrad_wolfram_teaching_kids_real_math_with
_computers.html
. In this talk he outlines his thoughts on how to use computers to implement a major change to our current math education system.

Math Cognitive Development and Rote Memorization

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.


References

Butterworth, Brian (2005). Dyslexia and dyscalculia: A review and programme of research. Retrieved 6/1/2012 from http://www.mathematicalbrain.com/pdf/THEORIES.PDF.

CCSS (2012). Common Core State Standards for mathematics. Retrieved 6/1/2012 from http://www.corestandards.org/the-standards/mathematics.

Devlin, Keith (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. Basic Books.

McLeod, Paul (2009). Jean Piaget. Retrieved 6/1/2012 from http://www.simplypsychology.org/piaget.html.

Moursund, David (2004, 2012). Using brain/mind science and computers to improve elementary school math education. Eugene, OR: Information Age Education. Access the PDF file at: http://i-a-e.org/downloads/doc_download/232-improving-elementary-school-math
-education-some-roles-of-brainmind-science-and-computers.html
.
Access the Microsoft Word file at: http://i-a-e.org/downloads/doc_download/231-improving-elementary-school-math
-education-some-roles-of-brainmind-science-and-computers.html
.

Wolfram, Conrad (2010). Teaching kids real math with computers. Retrieved 6/11/2012 from http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with
_computers.html
.

Van Hiele Model (n.d.). Accessed 6/1/2012 from http://en.wikipedia.org/wiki/Van_Hiele_model.


David Moursund

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.

For more information about David Moursund, see http://iae-pedia.org/David_Moursund.


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