Tuesday, February 5, 2013

A Genealogy of Realistic Mathematics Education

Time sinks are curious things. Some are tedious, some are frustrating, and some turn out to be fun. A few months ago, when I probably should have been studying for my comprehensive exams, a simple conversation in the office with +Ryan Grover started a genealogical journey (academically speaking) to trace back our origins and those of Realistic Mathematics Education (RME). I'm pretty good with my mathematics education history, and I knew RME in the United States took hold with the Mathematics in Context curriculum project, bringing Thomas Romberg and others at the University of Wisconsin together with Jan de Lange and others at the Freudenthal Institute at the University of Utrecht in the Netherlands. I suggested to Ryan that we search the Mathematics Genealogy Project to see if there were other ties between U.S. math ed and the history of RME, and the time sucking began in earnest.

With Ryan searching at his desk and me at the chalkboard, after several (or 4? 5?) hours we had traced back our academic roots many generations. Here's a glimpse of that work after some un-criss-crossing of arrows by Ryan, but probably still with a few mistakes:


Ryan and I left the office after dark. When I got home, my thoughts were still consumed about organizing and preserving this history, so I stayed up most of the night creating this in Google Drawings (part of Drive/Docs), where it would be easier to edit and be more shareable:
Years indicate when doctorate(s) earned. Download large version.
I've highlighted a few individuals who I think stand out. At the top left is Nicolaus Copernicus. We could have traced back a few more generations, but beginning with the person who is famous for not putting the Earth at the center of the universe seemed like a good place to start. The next one down, in yellow, is Jakob Thomasius. Although more of a philosopher, he was advised by Friedrich Leibniz and advised his more famous son, Gottfried Leibniz, and represents an early connection between the left and right sides of the diagram. The next person down, again in yellow, is Abraham Kästner. Ryan and I had never heard of him, but he's an extraordinarily connected fellow in this chart. Five of Kästner's 10 documented students are represented here, and an unseen one, to Johann Bartels, leads directly to Nikolai Lobachevsky. Furthermore, Kästner's bio on Wikipedia reads like some stereotypically tragic mathematician's drama, having been engaged to a woman for 12 years, only to marry her and see her die within the year. So then he had a daughter with his maid and spent his later years writing poetry.

If you look around, you'll find Kant, Euler, Gauss, and others, but prominently representing an early attention to mathematics education is Felix Klein, again in yellow. Klein became interested in the teaching of mathematics around 1900, and the International Commission on Mathematical Instruction (ICMI) has named their lifetime achievement award after Klein. From Klein we establish three major lines: the U.S. line through William Edward Story, a separate U.S. line to Maxime Bôcher, and a German line through Hilbert and Bieberbach. Curiously, the tree artwork on the main page of the Mathematics Genealogy Project shows the link from Klein to Story, although neither Klein's or Story's page establish a recognized advisor-advisee relationship. After checking a few other sources, it seems that making the Klein-Story connection is typical.

Now we have three major figures in the third row from the bottom. All were trained as mathematicians but transformed themselves into prominent figures and researchers in mathematics education. On the left is Ed Begle, colored in red to reflect his association with Stanford. Begle was the director of the School Mathematics Study Group, creators of what most call the "New Math" of the 1960s and 1970s. I don't want to overgeneralize, but Begle and his descendents tend to focus on the curriculum, instruction, and policy aspects of mathematics education.

Next is Henry Van Engen, colored in purple to signify his association with Iowa State Teachers College, now known as the University of Northern Iowa (my alma mater). Van Engen's publications going back to the 1940s reveal that he was focused on learning and meaning in mathematics. Instead of relying wholly on his mathematics background, he incorporated ideas from figures like Brownell and Piaget. Van Engen left ISTC in the late 1950s to help establish the math ed program at the University of Wisconsin - Madison, and his lineage of Leslie Steffe and Paul Cobb represent one of the strongest learning science traditions in math education.

On the right is Hans Freudenthal, shown in orange to signify his place in the Netherlands. A giant figure internationally, ICMI's other major international mathematics education award is named for Freudenthal in recognition of a major cumulative program of research. Whereas I associate Begle with curriculum, and Van Engen with learning science, to me Freudenthal represents a math education visionary and philosopher, someone able to reflect broadly on the field and history of mathematics and structure a new approach to mathematics education. Interestingly, Freudenthal's involvement in mathematics education was in part inspired by Begle and the New Math -- not liking what he saw in the New Math and fearing Europe would adopt a similar approach, Freudenthal steered the Netherlands in the direction we now call RME.

There are a few hidden connections at the bottom of the diagram that reflect my experience studying RME. Being at the Freudenthal Institute US and working with David Webb is the most prominent, but I greatly anticipate opportunities to learn from our FI colleagues from the Netherlands. Paul Cobb's collaboration with Koeno Gravemeijer in the early 2000s was mutually beneficial and has influenced me greatly, as Cobb's theories of learning mathematics work well in the context of RME.

Copernicus, more than 20 generations away, tends to be less influential.



Notes: Along the way I explored a number of other connections less related to RME and my perspective of it. William Brownell, for example, can be traced back through a number of psychologists to Kastner. Alan Shoenfeld, another mathematician-turned-math educator, can also be traced back to Kastner and has two different lines back to Gauss. Kastner seemed to turn up everywhere, while Issac Newton turned up nowhere.