What makes a good selfie -- By Axelle Faughn

Why use selfies to introduce and discuss mathematical concepts? What makes the idea so appealing to teachers and students alike? Obviously the power of visualization has a lot to do with it, turning abstract notions into concrete representations. In particular memory and recollection can be enhanced by the support of visuals when learning new material. Indeed, visual encoding is one technique our brain uses to commit new knowledge to long term memory through the process of storing new information by converting it into mental pictures. By systematically encouraging students to find visual representations of mathematical concepts in their daily world, we broaden the context of learning and create cues for retrieving such information outside of the academic classroom. We also provide opportunities for consolidation by unveiling a side of mathematics that may be more interesting, memorable, and engaging.

In the book "Teaching Mathematics as Storytelling", the authors dedicate a whole chapter to features of a good story. In this post I propose to attempt a similar analysis of components that enhance the use of selfies in the mathematics classroom. Some we have already discussed and may feel like a summary of good practice, others are new here.

The context of the photograph either brings the observer home (see "Derivatives in the Dorm" by Kathy), capitalizing on the familiarity of situations to better own the mathematical concepts, or it takes the observer on an adventure (as in Kenya), opening up new horizons for the group. Therefore, when selecting a selfie, it is important to be mindful of contextual associations, and mindful of the type of transfers one has to make when shifting one's attention from one context to another. The more representations of a specific mathematical concept students are exposed to, the more flexibility they have in making that shift efficiently.

"A part of good teaching that helps the transition to a richer understanding of mathematics is locating something wonderful in everything we teach" (p.18 of "Teaching Mathematics as Story-telling"). As most of us teaching mathematics know, elegance and beauty are not always conveyed as a easily to a group of students taking a pre-calculus class as they may be to an audience composed of mathematics aficionados. Instilling a sense of wonder (whether the term wonder reflects curiosity towards, or awe with respect to, the subject matter) in every mathematics classroom is a primary goal which can be achieved through the use of mathematical selfies. Indeed the search for appropriate representations in the world of the student partly helps them wonder about where it is they will find a good visualization that might appeal both to themselves and their peers, as well as receive the teacher's approval, therefore validating the quality of their submission. The sense of awe that students express when they realize how much of what they learn in the classroom also has ramification in their world outside of class is an added bonus of working with selfies... as one geometry student told me once "I never noticed before that there were triangles in all these objects!"

This sense of wonder leads us to discuss how to help students find human meaning in mathematics, in other words remembering that mathematics is a human construct, designed for human purpose. It is interesting to notice that students will either find a selfie expression in their world as in the mountain here, or they feel the need to make one up if they find themselves unable to retrieve an image from pre-existing objects (using belts or charging cords). For instance, submissions for polynomial functions often take one of these 2 forms.

Polynomial belt

In "Mathematical Selfies with an Artist's Perspective", Kathy follows one of her students on an artistic journey through the world of mathematical selfies. This aspect of selfies is highly motivating as a student engagement strategy and is further described as a possible multi- disciplinary approach in "STEAMing away with Mathematical Selfies", focusing on the artistic features of the students' work, yet providing additional human meaning to the mathematics. Likewise, using natural features to represent mathematical conventions (such as a natural set of axes for instance), or adding them in (drawing in) helps create a bridge with the mathematical world of the classroom. Students often use these spontaneously when trying to add an explanation within the image selected as shown in our post "A note on linear functions".


Selfies that offer a multi-approach, in other words those with a variety of entry points that can be used to illustrate more than one concept, are very rich mathematically and trigger long-lasting discussions in the classroom when properly navigated because of the mathematical connections they may reveal. These allow genuine student engagement during whole class discussions when students are invited to argue the validity of their interpretations. We discussed some of these in our most recent post "The problem with Number Sense", as well as  in "Mathematical connections unveiled within selfies".

The meta-selfie, or textual explanation that accompanies the photograph, helps shed light on the author's intent (see post "Reality and Perception" by Kathy for considerations on what the student's words may add to a picture), sometimes also adding extraneous information that may need to be addressed or clarified in class, opening up a world of opportunities for classroom discussions, as in the picture of the girl and the dog above where the student tried to use trigonometry before it was discussed in the corresponding course. In spite of the very approximate use of mathematics, the picture was voted one of the best by students taking the class, certainly due to the commonality of the situation and the humor factor created by establishing a seemingly complicated problem that could certainly be solved in a much easier way than the one suggested by the student (an age-old joke in school mathematics, or so it seems).

Reflection Selfie

To continue with the Humor theme, here is another student favorite that was voted best by the class when asked to submit pictures of transformations of functions. Although the notion of function itself is somewhat obsolete in this example, the physical reflection in the mirror allowed a good laugh, and helped capture students' attention, reminding us that fun can be had while doing mathematics. To that extent, any conflict, surprise, entertainment factor that personalizes a selfie and creates a feeling of familiarity with the mathematics will be favored by other students. Often peers will vote for selfies that I, as a teacher, would not necessarily have elected as most valuable. However, this reminds me that we do not approach the mathematics classroom from the same perspective, and selfies provide a wonderful way for us to meet in the middle.

Focusing on good features of a selfie, one can turn students interests from the extrinsic factors of obtaining good grades by performing the expected work, to more creative outlets and a more rewarding way to get involved in their learning, providing the intrinsic motivation that is often hard to gather in mathematics. Overall, students who are intrinsically motivated do better and understand the material more thoroughly, leading to creativity, high-quality learning, and self-efficacy. There is actual evidence that intrinsic motivation is a better predictor to success in mathematics than IQ measures (Middleton & Spanias, 1999). This alone should be convincing enough: students need to see math as interesting and useful, and it is our role as educators to convey such belief.

References:

James A. Middleton and Photini A. Spanias, "Motivation for Achievement in Mathematics: Findings, Generalizations, and Criticisms of the Research," Journal for Research on Mathematics Education 30, no. 1 (1999): 66.

Rina Zazkis and Peter Liljedahl, "Teaching Mathematics as Storytelling", SensePublishers 2009.