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America.
Summarizing 120+ Years of Research
in Math Education
David
Moursund
Professor Emeritus, College of Education
University of Oregon
My doctorate was in mathematics, and I have long been interested
in math education. My math education interest expanded to computers in
math education and then to computers in all of education. In this
newsletter I will use the term Information and Communication Technology
(ICT) to cover the full range of computers, calculators, computerized
devices, and the field of computer and information science.
If you have any involvement in math education, especially as a teacher,
parent, or teacher of math teachers, you will want to spend some time
browsing Alan Schoenfeld’s article, Research in Mathematics Education
(Schoenfeld, 12/22/2016). Schoenfeld is a world class math educator.
The following is a brief introduction to his professional career
(University of California, Berkeley, 2017):
[Schoenfeld] holds the International
Commission on Mathematics Instruction’s Klein Medal, the highest
international distinction in mathematics education; AERA's
Distinguished Contributions to Research in Education award, AERA’s
highest honor; and the Mathematical Association of America’s Mary P.
Dolciani award, given to a pure or applied mathematician for
distinguished contributions to the mathematical education of K-16
students.
Schoenfeld’s long paper cited above provides a history of math
education. The following are some brief quotations from his paper that
especially caught my attention.
Turn of the 20^{th} Century
Toward the turn of the 20th century,
mass education was an elementary affair, focused for the most part on
the three Rs. In 1890, only 6.7% of the 14-year-olds in the United
States attended high school, and only 3.5% of the 17-year-olds
graduated (Stanic, 1987, p. 150). The vast majority of schoolchildren
studied arithmetic with a practical bent: The main focus of instruction
was mastering arithmetic operations for the commercial marketplace. In
contrast, the small fraction of the population that enrolled in high
school (often en route to college) took courses in algebra, geometry,
and physics.
Wow! Look how far education in the United States has come in the past
126 years. We now have a high school graduation rate of over 80%, and
most of these graduates have taken at least three years of high school
mathematics.
Mid 1920s
Continuing to quote from Schoenfeld (2016):
By 1918, all states had compulsory
school attendance laws (Lleras-Muney, 9/19/2001). Of course, years of
required education varied from state to state. But, the math education
component of the 3Rs was well entrenched in the curriculum, and there
was considerable emphasis on rote memory and learning to make practical
use of math.
By 1926, mathematics curricular foci had shifted away from the abstract
topics (e.g., methods of solving mixture problems, greatest common
divisor, and Gregorian and Julian calendars) to the concrete—the
arithmetic of home and store, of maintaining a simple bank account, of
balancing a check-book, and other practically oriented applications.
1940s to 1960s. Beginnings of Learning for Understanding
Gradually there was a shift from a focus on practical uses of
mathematics toward students learning for understanding and problem
solving. This shift encountered considerable opposition from “back to
basics” groups. The idea of multiple learning theories was addressed in
1951 by Guy Buswell. Quoting Buswell in Schoenfeld’s article:
The very reason that there are
conflicting theories of learning is that some theories seem to afford a
better explanation of certain aspects or types of learning, while other
theories stress the application of pertinent evidence or accepted
principles to other aspects and types of learning. It should be
remembered that the factual data on which all theories must be based
are the same and equally accessible to all psychologists. Theories grow
and are popularized because of their particular value in explaining the
facts, but they are not always applied with equal emphasis to the whole
range of facts.
Thus, the “math wars” of the 1990s had a quite early beginning, and
still continue. Today, however, more and better research is gradually
leading to greater acceptance of students learning for understanding
and problem solving, with much less emphasis on rote memory (Moursund,
2016a).
Changing Nature of Math Education Research
Metacognition—thinking about one’s thinking—was a somewhat novel
concept in math education in the 1980s. It, and more recent research on
belief systems, are now important aspects of math education research.
See the Learning Theories website for a discussion of Carol Dweck’s
work on self-beliefs and learning (Learning Theories, 2017). Continuing
to quote from Schoenfeld (2016):
This work, as well as the work on
teaching and learning environments described below, is indicative of
the fact that the field has now reached the stage where there is a
fundamental and productive dialectic between theory and practice.
Research is no longer typically conducted in the laboratory and then
“applied” in classrooms. Rather, given that there are now tools for
reliable naturalistic observations and programmatic interventions,
classrooms can serve as laboratories. Research and development in
mathematics education increasingly live in powerful synergy.
All educational research faces the challenge of translating
research-based theory into effective classroom practice. Certainly math
education has faced and continues to face this challenge. Moreover,
math education and all other disciplines of study face the new
challenge of making effective use of Information and Communication
Technology (ICT). Here, the pace of change of technology is so fast
that our schools are falling further and further behind.
Information and Communication Technology in Education
Continuing to quote from Schoenfeld (2016):
Last but not least, technology. Here,
the story is not as clear, or as positive. The challenge in pragmatic
terms is that technological change comes so rapidly that it is
difficult for the research community (and practitioners!) to keep pace.
…
The iPad [tablet computer] was introduced in January 2010, and the
commercial world moves at much greater speed than either the research
community or the schools. An Education Week column by Michelle Davis
(2013) indicated the radical transformations taking place just 3 years
after tablets entered the marketplace.
Tablet computers are quite useful in some aspects of education.
However, the lack of a keyboard and the modest screen size are major
disadvantages when such tablets are compared to a laptop computer or a
computer with a still larger screen. Indeed, a display screen large
enough to display two documents side by side is a great aid to
productivity.
I have expressed my views on the importance of effective instructional uses of ICT in my new book, The Fourth R. This new fourth R is Reasoning/Computational
Thinking, learning to use human and computer brains together to help
represent and solve problems. The book recommends thoroughly
integrating the fourth R of
computer use into curriculum content, instructional processes, and
assessment starting at the earliest grades (Moursund, 12/23/2016).
Some Recent Thoughts of Larry Cuban
Larry Cuban is an emeritus professor in the Graduate School of
Education at Stanford University. He has written eloquently and
skeptically about computer uses in education for about 30 years. I have
always enjoyed his writing.
In essence, over and over again Cuban has said, “Show me the evidence.”
From his viewpoint, the research being done in in this field was not
meeting his standards of good, solid educational research.
This type of “show me the evidence” position is consistent with the
history of math education summary provided by Alan Schoenfeld. Math
researchers have a deep understanding of “rigorous proof” in
mathematics, and they tend to view the world in terms of the types of
thinking and careful proof needed to be successful as a mathematician.
Schoenfeld, for example, discusses the math wars from the point of view
that people on both sides were arguing from a lack of solid research
evidence. His position is that as the amount and quality of research
increases in this area, the issue “war” will slowly end.
I was somewhat amused and pleasantly surprised when I read Benjamin
Herold’s recent article, Ed-tech Skeptic Larry Cuban Finds New
Perspective (Herold, 2/7/2017). During the past year, Larry Cuban has
been visiting schools and talking with a number of people who are
involved in developing and implementing effective uses of computers in
education. In Herold’s article, Cuban responded to the question, “What
has made you less skeptical?” Cuban replied:
What I saw impressed me greatly.
Teachers are regularly and easily integrating technology. It's now in
the background, as common as paper and pencils and blackboards were
decades ago. The fact that it's moved from the foreground to the
background, to, "What are the learning goals of this lesson?" and "When
and how can I use these technologies to best achieve those goals?"—I
saw that going on and was very impressed.
I was pleased to read Cuban’s statement. It is consistent with my insights into the fourth R and the importance of thoroughly integrating ICT into the curriculum.
Here is Cuban’s comment about Summit, California, public schools:
Summit goes beyond a particular teacher
in a particular classroom. This was a whole school plan to integrate
technology and put it in the background, not the foreground. I saw how
the y develop a school culture, hire teachers, socialize everyone
working in that system to have that kind of direction. And it's in a
public school. I saw that at work and I was impressed with it.
…
There are three basic activities that teachers use in any class:
whole-group instruction, small-group instruction, and independent work.
What I saw at Summit, compared to other schools, was less whole group,
and more small group and more independent. And the technology helped
with that small group and independent learning.
[Here] I saw kids assessing how far along they were on goals they had
set. That's unusual and hard to do in a group of 25 to 30 with one
teacher. I saw technology as making that more possible in the hands of
an expert teacher.
Notice that this report from Larry Cuban does not represent carefully
done, replicable research. Rather, it represents his observations of
some current uses of computers in education that are consistent with
and supportive of his views of good schooling.
Final Remarks
I believe Schoenfeld has done a brilliant job in summarizing key
aspects of research in math education. However, I must admit
disappointment in Schoenfeld’s brief comments about computer
technology. I have been working in the field of ICT in education for
more than 50 years, and I was by no means the first person to be
working in the field (Moursund, 2016b). During that time, I had 76
doctoral students complete their PhDs at the University of Oregon in
various aspects of the field of ICT in education.
There has been substantial research on the use of computer-assisted
instructional materials in many different disciplines. Such materials
are gradually being improved and have been proven effective in many
different settings. Moreover, the use of Massive Open Online Courses
and other forms of online education are now having a significant impact
on both precollege and higher education (Moursund, 12/30/2015).
There has been substantial research on the effectiveness of students
learning to make use of ICT as an aid to solving the types of math
problems addressed in the “conventional” math curriculum. A crucial
question that I like to raise in this area is:
If a computer can solve or greatly help
in solving a type of math problem that is studied or could be studied
in the K-12 math curriculum, what can and/or should students be
learning about by-hand, by-computer, and jointly by-hand and
by-computer methods for solving this type of problem?
This is a very challenging question. Among other things, with the aid
of computers it becomes easier to add content to, and make changes in,
the sequencing of the K-12 math curriculum. Some ideas about this,
going back to Jim Fey’s research in the mid 1980s, are discussed in the
IAE Newsletter, Learning to Do and Doing to Learn (Moursund, January, 2017).
For years, math educators have made use of “think out loud” methodology
in which students are asked to verbalize the thinking they are doing as
they attack a math problem.
Now we have a variation on that. It consists of capturing every
keystroke as a student uses a computer to solve a problem or complete a
task. This is being used in many on-line courses, where a careful
analysis of this data is now contributing to improving such courses.
And, of course, a fruitful research methodology consists of using both
the think-out-loud and keystroke capture at the same time.
What You Can Do
Math education is certainly a researchable field. Schoenfeld’s
article shows the remarkable progress that has occurred and continues
to occur in both the theory and practice of math education.
I believe that every person who is involved in helping children to
learn math needs to focus some serious attention on the use of ICT as
an aid to instruction and as an aid to solving problems. It is crucial
that we help children learn to make effective use of the steadily
growing capabilities of ICT as an aid to solving problems throughout
the curriculum.
Teachers and parents, working at a grass roots level, can bring
considerable pressure to bear for changes in our conventional math
education system.
David
Moursund is an Emeritus Professor of Education at the University
of Oregon, and editor of the IAE
Newsletter.
His professional career includes founding the International Society for
Technology in Education (ISTE) in 1979, serving as ISTE’s executive
officer for 19 years, and establishing ISTE’s flagship publication, Learning and Leading with Technology.
He was the major professor or co-major professor for 82 doctoral
students. He has presented hundreds of professional talks and
workshops. He has authored or coauthored more than 60 academic books
and hundreds of articles. Many of these books are available free
online. See http://iaepedia.org/David_Moursund_Books.
In 2007, Moursund founded Information Age Education (IAE). IAE provides
free online educational materials via its IAE-pedia, IAE Newsletter, IAE Blog, and books. See http://iaepedia.org/Main_Page#IAE_in_a_Nutshell.
Information Age Education is now fully integrated into the 501(C)(3)
non-profit corporation, Advancement of Globally Appropriate Technology
and Education (AGATE) that was established in 2016. David Moursund is
the Chief Executuve Officer of AGATE.
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