Last updated on August 24, 2015
The Object-Process Duality in Mathematics
Several constructivist learning theories propose that a concept in mathematics may be perceived of as both an object and a process. For example, students may first think of functions as the process of evaluating a formula, but later recognize functions as objects that can themselves be manipulated in other processes. You might compose two functions to transform them into a new function, or you might explore properties of these mathematical objects such as their domain and range.
This object-process duality is a principle concept in APOS theory, a mathematical learning theory developed by Ed Dubinski and his colleagues at RUMEC. A similar idea arises in the work of Anna Sfard, and Gray and Tall’s theory of procepts. John Mason has described this dichotomy as a “delicate shift of attention,” and the some authors have suggested that the distinction in RME between a “model-of” versus a “model-for” represents a sort of object-process duality. The object-process duality also crops up in category theory, which is not a learning theory but an active area of mathematical research, and indeed, similar ideas also come up in Lakoff and Nuñez’s cognitive science of mathematics. This is not even a conclusive list of all the theories incorporating an object-process duality.
Such a widespread idea ought to have a neurological basis, and so I have been studying the object-process duality through the lens of neuroscience. The reality is that we do not understand the human brain well enough to draw empirical conclusions about the mechanics behind something like APOS theory. Instead, neuroscience offers a new way to frame and critique these ideas. This new perspective might help explain how students on the spectrum or students with ADHD encounter this dichotomy differently. It could provide a way to hash out differences in the aforementioned learning theories, and could potentially suggest new pedagogical treatments for getting students to see a concept as an object rather than a process.
I also want to better understand how a student’s prior knowledge of a concept might obstruct their ability to see this concept as an object rather than a process. I teach at a community college and my students have already encountered most of the topics we teach in another context. They may have developed a distaste for mathematics through years of bad experiences, and so they are not “constructing” knowledge from a clean slate. Tall has already explored this problem through his theory of met-befores, but no one has really explained how these met-befores might prescribe different pedagogical treatments for different students. This is one idea that I’m exploring in concert with the neuroscience research mentioned above.
Open Educational Technologies and Online Learning
I’ve been a Linux user for over a decade, and although I’m weary of clicky-clicky bling-bling, I’m excited to experiment with technology as a learning tool. I’m not interested in power points presentations. I’m interested in putting technology in the hands of our students, and letting them hack, create, and play in the process of learning.
In this TED talk, Margaret Wertheim of the Institute for Figuring explains how frilly crocheted objects and coral reefs physically model hyperbolic geometry. By playing with tangible objects such as crochet, students can understand difficult, abstract mathematical concepts. This idea inspired me to learn html5 canvas and start writing interactive web apps that my students can use to literally play with different mathematical concepts. For example, this web app, which I wrote, allows students to physically transform graphs themselves and see how each transformation affects the graph’s formula. Shortly after I wrote this web app, I was turned on to GeoGebra, which is like a graphing calculator and a platform for developing web apps in one. I’ve been using GeoGebra in my classes ever since.
I’ve also been very inspired by the ideas of Jim Groom and Gardner Campbell, among others, who have created entire classes by pasting together blogs, twitter accounts, and other technologies from around the web. These open classes have sometimes been dubbed “edupunk,” although Gardner Campbell does not like that term. I’m interested applying the edupunk approach could to mathematics classes. The goal is to create an open, online learning community where students can discuss and learn mathematics in an authentic, engaging environment. This is especially important work for so called “developmental” mathematics courses where MOOCs have clearly not lived up to their hype.
Flexibility and Creativity in Problem Solving
Writing calls on two skills that are so different that they usually conflict with each other: creating and criticizing. In other words, writing calls on the ability to create words and ideas out of yourself, but it also calls on the ability to criticize them in order to decide which ones to use. […] Most of the time it helps to separate the creating and criticizing processes so they don’t interfere with each other.
A similar phenomenon occurs in mathematics wherein students question their own ability to solve a problem before they’ve even given their own ideas a chance. In his seminal paper, What’s All the Fuss About Metacognition, Alan Schoenfeld observed that students would usually only attempt a single metacognitive strategy on a problem before running out of time or giving up. On the other hand, an expert problem solver would pivot quickly between different strategies until they’d found just the right one.
Our over-emphasis in mathematics on finding “the answer” comes at the expense of divergent thinking skills, which leads to the inflexible zombie problem-solvers that lurk classrooms everywhere. There’s a rich literature on problem-solving, dating at least back to Pólya’s famous How to Solve It book, but the quantitative and qualitative assessments of these approaches haven’t been too promising.
Inspired by Rothstein and Santana’s book, Make Just One Change, and Andrew Blair’s inquiry maths, I tried adopting an inquiry approach to problem solving in my quantitative reasoning class. Students were given a prompt and asked to generate questions about the prompt through a highly scaffolded process. They were then provided with a list of problem-solving heuristics, mostly pulled from Pólya’s book, and asked to select the approach they would take. An iterative design was built into the process so that students listed their chosen heuristic on a spiral, and moved around the spiral when they got stuck, reflecting on why the approach did not work and then selecting a new heuristic.
In parody of Peter Elbow’s “process writing,” my colleague and I deemed this approach “process mathing,” but it was not a successful approach. The questions generated from the prompt felt arbitrary and without direction, and the students loathed the overhead of keeping track of each metacognitive decision along the spiral. Most importantly though, it occurred to me that writing is different from “mathing” because it describes both the process of writing and the resulting artifact. On the other hand, there is no artifact in the problem-solving process.
In a paper by Bransford and colleagues, they describe an intervention built around the Jasper Woodbury Problem Solving Series where students construct different charts or graphs (described as “tools”) for the purpose of solving a problem. This shifts the focus from finding “the answer” to creating one of many tools that will get the job done.
Right now, I’m developing “process mathing 2.0,” which would involve students using an iterative design process to construct artifacts in order to solve a problem. The iterative design process would be divided into two steps: creating and criticizing. In the creation stage, students would employ techniques like Elbow’s “freewriting” to generate lots of ideas for the artifact, and in the criticizing stage they would narrow these ideas down to whatever is most useful for solving the problem. If their artifact cannot solve the problem then they would reflect on why and restart the process. The goal is to create and study an intervention that would empower students with better problem-solving skills so that they could learn math more independently and with greater self-confidence.