A Fractal Thinker Looks at Learning, Observing and Assessment

Edward B. Nuhfer, Idaho State University (nuhfed@isu.edu)

I wear two hats in academe. For the past dozen years, my larger hat has been as a faculty developer involved with the primary mission of helping faculty succeed in their professions. My smaller hat, still worn daily, is as a professor of geology. My thinking about what constitutes effective development and assessment is influenced by my discipline—geology—in particular, geologists' understanding of the shapes of natural features and patterns of events in time. Until recently, we used to refer to such shapes and patterns as "random" or "complex." Geologists have since learned that such features are not, in fact, random. Rather, many possess an order now termed "fractal." I have found that perceiving education and development as fractal forms and components provides a vision extremely useful for practice.

"Y" Fractals?

Consider a tree or the branching network of neural connections in the brain. Branching networks look complex, but such forms use a basic building block similar in shape to the letter "Y." If you replace the upper two branches of the letter "Y" with other "Y's," and so forth, such replacement is a recursive operation. Fractal features appear indescribably complex until one sees them produced by a recursive repetition of some very basic building block called a generator. A generator (such as a "Y") is a form made from more basic Euclidean pieces, such as a straight line. In fractal language, the simple pieces that make a generator are called initiators. A characteristic of any fractal pattern is that it requires recursive use of a basic generator to build it. Fractal forms differ from Euclidean forms because the latter (squares, polygons, trapezoids) don't require any recursive operation and can be made merely by extending and changing the direction of a continuous straight line. But a fractal form requires a generator, not just an initiator, and the recursive operation uses the entire generator.

Figure 1. Concept of a fractal branching form.

In education, single practices or pedagogies are analogous to initiators, not generators. One cannot build successful education through a focus on pedagogical methods alone, no matter how much one believes in their efficacy. If faculty believe, for example, "If we master collaborative learning methods, all will be well," they have put far too much faith in a single pedagogical technique. A truly dynamic basis for faculty development (and education in general) grows from a more comprehensive, applied philosophy.

A trait of a fractal form is that its dimensions change depending upon the length of the measuring instrument one uses to measure them. Our common sense is trained by experience to think in Euclidean terms. Rectangles and triangles don't yield a change in perimeter based on the tool we use. The fact that something would change in length depending upon the length of our measuring stick defies our common sense, but that is exactly how fractals behave. For instance, take the profile of a coastline on a large map. If you measure its length with a pair of dividers that are set initially at say, four inches, and then again, at two inches, one, inch, a half-inch and so forth, each measured length of the coastline will increase, as the dividers get smaller. To some, it may seem obvious that if you take a very crooked line (like a coastline) and measure it crudely (Figure 2) and then measure it with increasing precision (Figure 3), its length will increase.

Figure 2. Measure a coastline by dividers set wide.

Figure 3. Measure a coastline by dividers set narrow.

However, the length of a fractal form just doesn't increase randomly, it increases with such regularity that one can predict accurately the length one will measure based upon any setting of the divider. This quantifiable increase (related to "fractal dimension") provides a concise descriptionóa name expressed in numbersóto distinguish one kind of coastline from another. If we are touring the coastline by boat, a fractal dimension cannot tell us what lies ahead around the bend, but if we take two trips along two coasts with different fractal dimensions, we can have confidence that we'll see differences we can describe.

Figure 4. Order of coastline length increase L as function of decreased divider setting r.

Fractals are not restricted to just natural material forms. Some patterns of events in time are also fractal. We can deduce recurrence intervals for floods (or rainfall) by graphing events of a certain magnitude at different time scales (a decade, a century, a thousand years). Such events are perceived through common sense as "random," but a plot of magnitude versus a function of frequency creates a pattern where, once again, the events fall closely along a straight line.

Figure 5. Floods and rainstorms are examples of fractal patterns in time.

Figure 6. The linear order present in a recurrence interval plot reflects fractal pattern in time of such events.

The slope of the line expresses fractal dimension. What we perceived intuitively as random reveals an order that lets us know how large a two-thousand year event might be as opposed to a one-hundred year event. It is important to realize that the pattern described does not permit us to predict when a given flood will occur. In fact, understanding the ways that large natural events in fractal time patterns are produced seems to indicate an impossibility for any such predictions.

Teaching's Coastline

The reference to a coastline or floods as "random" or "too complex to describe simply" reminds me of statements "Teaching is such a complex activity that we can't really describe good teaching." If it is impossible to measure or recognize something (such as good teaching or high quality education), it makes little sense to claim to be able to affect or improve it. How can one claim to assess or improve what one can't describe? This gets discomfortingly close to claims often made by pseudo-scientists and charlatans. However, I believe that the complexity we experience in education, which seems to grow the more we study it, has order inside the apparent chaos and that there is more than a metaphorical relationship between teaching, education, development and fractals.

The classic work of William G. Perry, confirmed repeatedly by later researchers, proves that becoming educated results from a pattern of experiences in time. These experiences are neither regular nor constant, but produce results that are neither random nor indescribable. This sounds much like the fractal pattern of events in time. The development of complex branching synaptic connections in the brain is the biological transformation that takes place because of becoming educated. Such patterns, like branching trees, are material fractal forms.

If the enterprise of education has a fractal nature, then a generator is involved in producing those patterns that lead to successful large-scale outcomes—the primary business of higher education. In a fractal, the small-scale features can indeed be recognized as strongly related to the larger scale ones. This has profound implications. The most important concept a faculty developer might master derives from distinguishing the nature of a generator that produces high quality education as opposed to one that produces outcomes of lesser quality. If, as developers, we can point faculty toward an operational generator that produces sophisticated, comprehensive, working philosophies, we can foster far better educational success across the institution than if we focus too much on fixes and techniques.

The branching "Y" is pertinent to learning because it roughly describes the development of neural networks that are the products of learning. Successful professional practice arises from learning, so in faculty development, it makes sense to think: "What kind of generator worked upon recursively over time will produce successful college instructors?" We worked on that very question through Boot Camp for Profs® for over a decade to deduce the generator shown in Figure 6. It has six components: (1) self-introspection, (2) content, (3) pedagogy (4) thinking, (5) rubrics, and (6) student self-assessment. Over time, we found that this integrated approach was more effective in producing permanent career success than were any series of workshops.

Figure 6. The successful educator's generator and its manifestations at varied scales.

The parts encompass three broad concepts: teaching and learning (which are often discussed) and thinking (which is an embarrassingly absent topic in many faculty development meetings). This approach to development provides faculty members with the training and support to build their own strong generators rather than simply master a few teaching tricks or even to achieve high student ratings. All six parts of the generator come with their own tools to assess successful outcomes, and all six parts can be applied appropriately at all five scales described above. A sophisticated philosophy can result only where the faculty member is actively aware of all parts of the "Y."

Suggesting that fractals serve as a useful way to perceive our role does not advocate for reducing education to a formula or for practicing any facet of it in a mechanical way. The base of the generator in Figure 1 is introspection, and this admits limitless affective variables such as intuition, emotional intelligence, caring, etc. The fractal model affirms that small changes anywhere in the generator will produce large effects on the complex form made from it.

A tour of the generator

 (1) Personal Introspection. An essential step in development lies in helping individual faculty members discover how and when their ideas and values were built. Personal introspection is the foundation of the generator—the trunk of the tree. I consider it the most important part of the generator, because if it is done sincerely, with some sophistication and persistence over time, it may lead the individual to acquire the other essential components of the generator. Through introspection a professor makes the fundamental decisions about what it is he or she most wants to do and why. When one isn't clearly striving to do what he most values, effort is spent producing things that one does not value. For example, every professor values good relationships with students. Suppose a new professor's experience involves having to deal with late submissions of papers or submissions of poor quality. The experience abrades the professor's spirit, and the following term she produces a syllabus that tries to prevent a recurrence by including guidelines unconsciously written in a scolding tone. It's possible to achieve orderly submissions without scolding new students who had nothing to do with the professor's initial bad experience, but it's likely that scolding guidelines will end up replicating that initial experience.

Because it is a form of learning or self-teaching, introspection is essential in the construction of the synaptic connections that enable good teaching. No learning or change in learning takes place without neural, physiological transformation: learning begins with what one believes at the moment. Initial introspection catalogs the current state of knowing, and this awareness can help one to direct and to chart progress in any needed different direction. The affective traits of this component are dominant and essential. The best instructors at an institution seem invariably to be those who actually do what they love.

A faculty member who has a clear view of what he or she wants to produce in the form of learning outcomes (content learning, pedagogical experiences, attitudes, levels of thinking) is ready for advancement through the support of a faculty developer or an accomplished peer. An assessable product of such introspection is an initial thoughtful teaching philosophy. As professors become more highly developed, the sophistication of introspection and the product of it will improve dramatically.

2. Content Learning. Faculty spend decades of their lives centered on the acquisition and use of particular content, so content knowledge is the portion of the generator that faculty relate to easily. Although developers often hear of the conflict between research and teaching, research is also tied intimately to content—contributing new content to the discipline. Although research is not closely tied to instructional success, most institutions consider it a desirable (even essential) component of professional success. Disciplines, linked in many cases to professions, have established learning communities of like-minded people who provide validation, respect, and support for those identified with achievements that are all associated with a particularly identifiable set of content knowledge. Where educational institutions fail to provide equivalent validation, respect and support, it's easy to see how faculty often develop closer identification with their disciplines than with their institutions. A problem between research and teaching arises when professors identify (or are forced to identify) so much with content knowledge that they have little regard for being with students who have not yet developed equivalent neural networks that enable appreciation, interest, or comprehension of the content—i.e., they aren't chemists, literary critics or psychologists yet, for heaven's sakes!

While most developers would probably prefer to begin with introspection, faculty may prefer that we begin with enhancing their ability to promote content learning. Developers need to understand that a focus on content offers a best choice as a starting point to begin professional development. The knowledge survey (Nuhfer and Knipp, 2003) is a tool that provides a way to assess content learning, directs attention to improved practice, and makes that content more visible to both professor and student. It meets the professor halfway. Making content choices quite visible enables one to move on to queries about "Why this content?" "How does this match the needs of students?" "What kinds of background are presumed?" and "What kind of pedagogical approaches might best convey the content?"

3. Pedagogy. Pedagogy refers to a broad range of strategies designed to produce experiences that enhance learning. Most faculty development workshops focus on pedagogical techniques, and one volume after another appears that advocates for a particular type of pedagogical strategy (and sometimes denigrates others, most often, lecture). The adamant advocacy of a single technique or approach ignores learning diversity (many students can indeed learn well through lecture-discussion approaches) and shuts the door on the possibility that: (a) pedagogy may be matched to the content being taught and (b) particular practices may be better matches for particular students. The idea of conveying carefully chosen content with particular pedagogical methods, matched with appropriate assessments, follows the concept of "instructional alignment" described by Cohen. Instructional alignment seems to be the only classroom method that can match tutoring with respect to achievement in content learning outcomes (Bloom 1984). As soon as one realizes that synapses are built through confronting material in diverse ways, it becomes obvious why an instructor must master more than one pedagogical technique.

Together, introspection and the resulting choices of content and choices of pedagogy are the primary basis for the profession that is often called "teaching." Although one hears that the emphasis should be on learning rather than teaching, teaching is the design of experiences that promote optimal learning. The two are inseparable in advancing any serious educational enterprise. Learning indeed can and does occur without any influence by a teacher, but such experiential learning in itself is limited. Such undirected learning could neither produce nor even support the levels of knowledge required by any modern society. Sophisticated learning in critical masses of citizens requires sophisticated teaching. Formative surveys are the assessment tools of pedagogy (for example, see Hildebrand, et al.). These provide a "fingerprint" of a professor's teaching style and obviate choices for improvement. Documenting the use of sound pedagogical strategies that research has proven useful to students' learning is probably more important in defining "good teaching" than collecting one's scores on global summative student ratings.

4. Levels of Thinking. Try this experiment in your own classes. Have students complete the following sentence: "I consider that I will have achieved a good education at X college/university if...." See how often an improved ability to think is mentioned as a perceived measure of success. I tried this recently in a class of ninety students. Not a single student referred to "thinking" in answering. In our flurry of discussions on "teaching and learning" or even in catchy titles with a high fog index "less teaching more learning," we don't talk much about thinking. As a result, the public (and our students!) still regard "becoming educated" as mastery of some content knowledge and the ability to obtain gainful employment as the most valued outcomes defining a successful undergraduate education. However, enhancing one's level of thinking capacity is surely the most valuable outcome a college experience can provide. Enhanced thinking capacity enables one not merely to get a job, but also to advance in a career and to take care of oneself by reasonably evaluating difficult, often ambiguous situations.

In two papers, we synthesized key research on levels of thinking (Nuhfer and Pavelich 2001, 2002). This research offers several revelations of importance to the fractal model. (1) High level thinking has the signature attribute of being able to effectively use evidence to confront open-ended problems. (2) High level thinking seldom arises as a spontaneous product of higher education unless deliberately designed into a degree program in a planned sequence. (3) Content mastery is necessary in order to achieve high level thinking; it does not develop separately from content. In order to use evidence of substance—that is to say, make decisions based on evidence—one must be able to understand the evidence in order to evaluate it. This requires one to have mastered the discourse and some reasoning frameworks of a discipline. Teaching and learning may be inseparable, but both produce content learning without moving students to higher levels of thinking. Therefore, a sophisticated awareness not only of teaching and learning, but also of thinking, must be part of any educational philosophy aimed at producing educated, rather than merely "trained" students.

5. Rubrics. It is easy to presume that when we challenge students at the higher levels of Bloom's cognitive taxonomy (Bloom 1956), we are teaching high-level thinking. We forget that Bloom's taxonomy is a teaching-centered scheme that evaluates the level of challenge, but not the level of students' responses. In contrast, schemes such as the Perry Model or the Reflective Judgment model evaluate students' performance. In order to teach students to recognize when they are meeting a high level challenge with high level performance, rubrics are essential. Rubrics are criteria that a professor discloses as a way to evaluate a response to an open-ended assignment such as a journal, a written assignment or a project. A rubric conveys a framework that a student should use to evaluate the quality of his work before submitting it. The term itself is relatively new, yet it is probably impossible for a professor to produce high-level learning experiences without finding a conscious, deliberate way to disclose such criteria as part of his teaching one way or another. Documenting success in producing high-level thinking requires assessment of students' performance. One efficient way to assess high-level thinking is to present representative high-level challenges, the rubrics used to evaluate these, and the students' products that result from meeting the requirements of the rubric. Rubrics are fairly new to higher education. The first book designed specifically for university level teachers (Stevens and Levi, 2005) was only released last month.

6. Student Self-Assessment. Student self-assessment is the students' equivalent of professors' introspection. Alverno College (2000) was the first to understand the essential value of self-assessment and remains the institution that has developed this understanding further than any other. Self-assessment does not rest with content learning but it extends metacognitive awareness throughout the learning process. This practice results in an accurate review by each student about how one's content knowledge, attitudes, skills, frameworks of reasoning and feelings were utilized and changed as a result of a learning experience. This review serves as the basis of a plan for how to improve one's performance the next time one encounters a similar challenge. The "Framework for Self Assessment" (Alverno College, 2000) is the rubric through which students and faculty evaluate progress from beginning through intermediate to advanced levels of reasoning. Within the fractal generator model presented here, an assessment product would be the self-assessment journal of the student.

Professors find that all six components of the generator are useful to development of ever more sophisticated practices, philosophies, and assessments that contribute to sharpened insight and skills. None of this is easy or formulaic. The work of developers is, as always, supporting, promoting and encouraging faculty in the arduous labor of building the comprehensive understanding needed to insure students' success.

The building of outcomes based on these components is analogous to the way a tree builds itself into a larger complex fractal form through repeated addition of branching patterns. Neural networks are also branching fractal forms, and these result from acts of learning. Sadly, this does not imply that what is learned is correct; learning that is incorrect also builds neural networks. Leamnson (1999) alludes to the problems this causes for students: they arrive with many ideas about how the world works which are dead wrong but difficult to dislodge.

Figure 7. Leamnson's (1999) view of learning in accord with the basic biology of synapses.

Lewis Wolpert (1992) notes the same problem with the most natural interpretations of experienced phenomena through "common sense." One must overcome such reliance on "common sense" before one can understand science clearly. The degree to which students' ideas accord with physical reality is likely to be the degree to which the student who owns them achieves success in science. Similarly, developers discover that every faculty member arrives with ideas and values about his or her teaching and the role it plays in students' learning. Generally, faculty turn to developers only when what once worked successfully seems to fail.

Figure 8. Analogy of challenges to assessment of learning from fractal neural connections. The brain develops synaptic connections that involve both cognitive growth (green) and affective factors. Global items such as those found on student evaluations sample general combinations of affective and cognitive on a very general basis. Knowledge surveys are more in accord with understanding a coastline--through multiple measures.

All six components plus the generator itself can be assessed through various tools and products

Table 1. Generator components with a short list (not exhaustive) of tools and products that can be assessed. The best overall assessment of the entire generator lies in a sophisticated teaching philosophy that shows awareness of all generator components. Evaluation of a faculty member can consist of looking for evidence that the sophisticated philosophy is consistent with practice.

The revolution of plate tectonics arose primarily because a new perception permitted us to reorganize available information on materials, processes and rates to make this all part of a more understandable system interrelated in space and time. Fractals may offer a similar way to view education and the role of a developer within an academic community. So taking a fractal view does not involve new information, and does not imply that education is anything other than complex. What fractals do for us is aid the perception of complexity in a way that adds clarity and informs practice.  

Final Note. This summary draft essay for NAGT is taken largely from a series of ongoing columns written for National Teaching and Learning Forum. Those wishing more detail can consult the articles that follow. Much of this work is copyrighted by National Teaching and Learning Forum or a work in progress under a book contract. The page can be freely consulted by NAGT and its representatives but it may not be rewritten or altered in any way without the author's written consent.

Nuhfer, E. B., 2003, Developing in fractal patterns I: Moving beyond diagnoses, evaluations and fixes: National Teaching and Learning Forum, v. 12, n. 2, pp. 7-9.

Nuhfer, E. B., 2003, Developing in fractal patterns II: A tour of the generator: National Teaching and Learning Forum, v. 12, n. 4, pp. 9-11.

Nuhfer, E. B., Krest, M., and Handelsman, M., 2003, Developing in Fractal Patterns III: A Guide for Composing Teaching Philosophies: National Teaching and Learning Forum, v. 12, n. 5, pp. 10-11.

Nuhfer, E. B., and Adkison, S., 2003, Developing in Fractal Patterns IV: Unit Level Development -Teaching Philosophies at the Unit Level: National Teaching and Learning Forum, v. 12, n. 6, pp. 4-7.

Nuhfer, E. B., 2003, Content Coverage, Courses, and Controversy Part 1: Developing in Fractal Patterns V: National Teaching and Learning Forum, v. 13, n. 1, pp. 8-10.

Nuhfer, E. B., Leonard, L., and Akersten, S., 2004, Content Coverage, Courses, and Controversy Part 2: Developing in Fractal Patterns VI: National Teaching and Learning Forum, v. 13, n. 2, pp. 8-11.

Nuhfer, E. B., 2004, Student Management Teams: Fractals for Students Too:Developing in Fractal Patterns VII: National Teaching and Learning Forum, v. 13, n. 4, pp. 7-11.

Nuhfer, E. B., 2004, Aikido, faculty development and teaching: Systems in Harmony: Educating in Fractal Patterns VIII: National Teaching and Learning Forum, v. 13, n. 5, pp. 9-11.

Nuhfer, E. B., 2004, Why Rubrics?: Educating in Fractal Patterns IX: National Teaching and Learning Forum, v. 13, n. 6, pp. 9-11.

Nuhfer, E. B., 2004, Fractal Thoughts on the Forbidden Affective in Teaching Evaluation & High Level Thinking: Educating in Fractal Patterns X: National Teaching and Learning Forum, v. 14, n. 1, pp. 9-11.

Nuhfer, E. B., 2005, Tests as anchors that wobble: Understanding imperfect correlations in educational measurements: Educating in Fractal Patterns XI: National Teaching and Learning Forum, v. 14, n. 2, pp. 8-11.

Nuhfer, E. B., and Pavelich, M., 2001, Levels of thinking and educational outcomes: National Teaching and Learning Forum, v. 11, n. 1, pp. 5-8.

Nuhfer, E. B., and Pavelich, M., 2002, Using what we know to promote high level thinking outcomes: National Teaching and Learning Forum, v. 11, n. 3, pp. 6-8

The example of the coastline comes from Benoit Mandelbrot's article in Science, 1967, "How long is the Coast of Britain?"

All the above are available online to institutional subscribers. The Boot Camp for Profs® faculty development program is designed in accord with the fractal model. Readers may find that program at the web site http://www.isu.edu/ctl/nutshells/old_nutshells/6_604.htm. In fact, enjoy all the Nutshells at http://www.isu.edu/ctl/nutshells/index.html.

 

References Cited

Alverno College Faculty. 2000. Self Assessment at Alverno College. G. Loacker, ed. Milwaukee, WI: Alverno College.

Bloom, B. S. 1956. Taxonomy of Educational Objectives:The Classification of Educational Goals. Handbook I. - Cognitive Domain. New York, NY: David McKay.

Bloom, B. S., 1984. "The 2 Sigma Problem: The Search for Methods of Group Instruction as Effective as One-to-one Tutoring." Educational Researcher 4(6), 4-16.

Cohen, S. A. 1987. "Instructional Alignment: Searching for a Magic Bullet." Educational Researcher 16(8), 16-20.

Hildebrand, M., Wilson, R. C., and Dienst, E. R. 1971. Evaluating University Teaching. University of California, Berkeley, Center for Research and Development in Higher Education.

King, P. M. and Kitchener, K. S. (1994). Developing Reflective Judgment. San Francisco, CA: Jossey-Bass.

Leamnson, R. 1999. Thinking About Teaching and Learning: Developing Habits of Learning with First Year College and University Students. Sterling, VA: Stylus.

Nuhfer, E. B., and Knipp, D. 2003. "The Knowledge Survey: A Tool for All Reasons." To Improve the Academy 21, 59-78.

Nuhfer, E. B., and Pavelich, M. 2001. "Levels of Thinking and Educational Outcomes." National Teaching and Learning Forum 11(1), 5-8.

Nuhfer, E. B., and Pavelich, M. 2001, 2002. "Using What We Know to Promote High Level Thinking Outcomes." National Teaching and Learning Forum 11(3), 6-8.

Perry, W. G. Jr. 1999. Forms of Ethical and Intellectual Development in the College Years. San Francisco, CA: Jossey-Bass (reprint of the original 1968 work with minor updating).

Stevens, D. D., and Levi, A. J.,2005, Introduction to Rubrics. Sterling, VA: Stylus.

Wolpert, L. 1992. The Unnatural Nature of Science. Cambridge, MA: Harvard University Press.