# Teaching¶

In my mathematics undergraduate studies I made one important experience: Homework assignments are an integer part of learning. Yet, it seems fairly obvious how appropriate mathematics assignments may look—”Prove theorem XX”, “Show that...”—and there is a lot of experience with respect to the level that is appropriate for different courses. Unfortunately, nothing like that is true for perceptual psychology. Typical psychology assignments involve summarizing classical results or presenting a circumscribed topic in class. The big difference to mathematical assignments is that students reproduce other people’s work rather than solve small problems. Clearly, psychology is an empirical science while mathematics is theoretical and simple problems can usually be solved using paper and pencil. Is there a way to have students solve problems in perceptual psychology assignments as well? I think yes. The following two sections present two examples.

## “Models of higher brain function”¶

At the Bernstein Center for Computational Neuroscience I have been tutoring a class in which students were to explore little problems from computational neuroscience using the Python programming language. For example, after learning the basics of natural image statistics, students computed power spectral densities of a number of natural images and applied MDP’s PCA and ICA to small collections of images.

As part of the same learning module, I gave two lectures with analytical assignments about the basic notions of natural image statistics. Here, students were to prove some basic statements about redundancy and redundancy reduction. I often feel that introductions to natural image statistics put too much emphasis on linear methods. Yet, it is fairly straight forward to prove some redundancy reducing properties of sensory gain control. Although the full model by Schwartz & Simoncelli [1] is analytically difficult to study, it is possible to reduce this model to a simple parametric form and study Schwartz & Simoncelli’s statements using elementary probability theory. This reduced problem was studied in that class.

## “Statistical models for data processing”¶

Not all psychology students learn programming, and even elementary
probability theory might be fairly technical for most psychology students.
Yet, all psychology students learn statistics. An integer part of
statistics—maybe the only purpose of it?—is modeling. Statistics is to a
very large extend about formulating models and testing these models on
empirical data. With Simon Barthelme, I held a block course on
statistical modeling at the Bernstein Center for Computational Neuroscience.
In this class, we explored a new type of assignments: students did little
perceptual experiments on themselves and applied the collected data. One
class of perceptual phenomena that proved very robust and well suited for
these experiments was *length perception*: The lengths of horizontal lines
are usually perceived shorter than the lengths of vertical lines.

There are several nice aspects about these exercises:

- They can be completed using paper and pencil. For example, we handed out assignment sheets with a collection of horizontal lines and asked students to add vertical lines they perceived to have the same length (i.e. without using a ruler).
- They can be formulated completely in the language of perception. For example, one may ask how much longer do you perceive the vertical line?
- The effects are strong enough that students only have to do very few “trials” and can mainly concentrate on translating the perceptual question into a statistical model.
- Many of the effects can already be studied with very basic statistical tools.

Exercises like this can be set up such that they emphasize statistical methods, which is what we did in that class. Alternatively, many fundamental ideas of reasoning about perception can be illustrated using line lengths. These include for example the differences between perception and physical reality, the quantification of perceptual effects in terms of equivalent physical effects, or effects from context and attention. In fact, it might be that the only thing that is difficult to present with (superthreshold) lines it that of an absolute threshold. Yet, there are good reasons to actually doubt the existence of absolute thresholds anyway (e.g. [2]). Thus, I believe that this limitation is less serious than it might sound first.

[1] | Schwartz, O and Simoncelli, EP (2001): Natural signal statistics and
sensory gain control. Nat Neurosci, 4(8), 819-825. |

[2] | Swets, JA (1961): Is there a sensory threshold?. Science, 134,
168-177. |