Information Age Education
   Issue Number 203
February 14, 2017   

This free Information Age Education Newsletter is edited by Dave Moursund and produced by Ken Loge. The newsletter is one component of the Information Age Education (IAE) publications. All of the IAE materials are free and can be accessed at http://iae-pedia.org/Main_Page.

All back issues of the newsletter and subscription information are available online. In addition, six free books based on the newsletters are available: Validity and Credibility of Information; Education for Students’ Futures; Understanding and Mastering Complexity; Consciousness and Morality: Recent Research Developments; Creating an Appropriate 21st Century Education; and Common Core State Standards for Education in 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 20th 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.

References and Resources

Herold, B. (2/7/2017). Ed-tech skeptic Larry Cuban finds new perspective. Education Week. Retrieved 2/10/2017 from http://www.edweek.org/ew/articles/2017/02/08/ed-tech-skeptic-larry-cuban-finds-
new-perspective.html?cmp=eml-enl-eu-news2-RM
.

Learning Theories (2017). Self-theories (Dweck). Retrieved 2/9/2017 from https://www.learning-theories.com/self-theories-dweck.html.

Lleras-Muney, A. (9/19/2001). Were compulsory attendance and child labor laws effective? An analysis from 1915 to 1939. Retrieved 2/9/2017 from http://www.econ.ucla.edu/alleras/papers/education%20paper-revised2.pdf.

Moursund, D. (January, 2017). Learning to do and doing to learn. IAE Newsletter. Retrieved 2/9/2017 from http://i-a-e.org/newsletters/IAE-Newsletter-2017-201.html.

Moursund, D. (12/23/2016). The Fourth R. Eugene, OR: Information Age Education. Download the Microsoft Word file from http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/289-the-fourth-r/file.html. Download the PDF file from http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/290-the-fourth-r-1/file.html. Access the book online at http://iae-pedia.org/The_Fourth_R.

Moursund, D. (2016a). Math education wars. IAE-pedia. Retrieved 2/9/2017 from http://iae-pedia.org/Math_Education_Wars.

Moursund, D. (2016b). History of computers in education. IAE-pedia. Retrieved 2/9/2017 from http://iae-pedia.org/History_of_Computers_in_Education.

Moursund, D. (12/30/2015). MOOC enrollment continues to grow. IAE Blog. Retrieved 2/10/2017 from http://i-a-e.org/iae-blog/entry/mooc-enrollment-continues-to-grow.html.

Schoenfeld, A.H. (12/22/2016). Chapter 14: Research in mathematics education. Sage Journals. PDF file retrieved 2/9/2017 from http://journals.sagepub.com/doi/pdf/10.3102/0091732X16658650.

University of California, Berkeley (2017). Alan H. Schoenfeld. Retrieved 2/9/2017 from https://gse.berkeley.edu/people/alan-h-schoenfeld.

Author

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.

Email: moursund@uoregon.edu.

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Information Age Education is a non-profit organization dedicated to improving education for learners of all ages throughout the world. Current IAE activities and free materials include the IAE-pedia at http://iae-pedia.org, a Website containing free books and articles at http://i-a-e.org/, a Blog at http://i-a-e.org/iae-blog.html, and the free newsletter you are now reading. See all back issues of the Blog at http://iae-pedia.org/IAE_Blog and all back issues of the Newsletter at http://i-a-e.org/iae-newsletter.html.