Classroom Activity: Math Selfie Treasure Hunt - by Axelle Faughn

One common concern when teaching mathematics is to provide relevance to the "real world" and to answer the recurring question "When am I ever going to use this?". Meeting this need can be addressed through activities that pertain to any of the following three categories:

• Math in the Student’s World of the Future:  Someday I’ll need math in my job as a _________.
• Math in the Student’s World of the Classroom:  My teacher is having us do this activity where we use some application of math.
• Math in the Student’s World of Now:  What does that mathematical concept, technique or definition look like?

By using mathematical selfies in the classroom we are clearly concerned with the latter. Not only do we explore mathematics in the student's world, but we use modes of communication prevalent in social media through "pictures & quotes", attempting to convey big ideas in a snapshot. In this post I describe some of the ways that we have used selfies with our students, and I invite comments on other possible activities that can be conducted in or outside of the classroom.

1. Mathematical Definitions Portfolio

Both Kathy and I used this one in geometry class, so students would create a book of definitions including every new term introduced in class. Without a common language and clear definitions of concepts, one would not be able to communicate in the math classroom. In order for ideas to be shared and understood students have to master the basic rules of syntax and discourse that are specific to the topic of study. Students were instructed to provide the standard definition of the term, along with a picture illustrating the concepts and an explanation for why the particular representation was an appropriate visual example.
Here is a possible entry from one student's portfolio, as an illustration that would accompany the more standard euclidean definition of two distinct lines that are always the same distance apart and never intersect. But why do blinds need to be parallel? Well... apart from considerations of appropriately directing or blocking light, we've all seen the mess it creates when they choose not to be!

2. Mathematical Selfies Treasure Hunt

This activity is very engaging and we have used it both with our students and for conference workshops. Here are the directions for conducting a Math Selfies Treasure Hunt:

Based on a set of mathematical topics, terms, concepts, or techniques, find photo opportunities using the ever available phone (including yourself is fun!) to illustrate what a particular mathematical item means to you or how you understand it. Collect all of the photos for each item for display... Ready, Set, Go!

* Find a team of 2 to 4, making sure you have at least one cell-phone or Ipad per team

* Choose a category:  Number sense, Geometry, Function (and you can certainly add others)

* Find and photograph an example of each concept, technique, or term on your list of mathematical treasures. Sample lists and submission examples are shown below.

* Email/upload photos to [include email address or link to goosechase.com] with a short title that identifies your mathematical selfie

Number Sense Selfie Hunt List

1.     Illustrate place value or base 10 notation

2.     Illustrate the Concept of number operations

3.     Illustrate Properties of number operations

4.     Illustrate the concepts of measurements and units

Sequence 1, 3, 5: skip-counting

5.     Illustrate geometric thinking

6.     Illustrate a number line

7.     Proportional reasoning

8.     Definition of Fractions

9.     Illustrate concept of fraction operations

10.  Decimal

11.  Percents

12.  Illustrate types of numbers
(whole numbers, integers,
rational, irrational, real)

Function Selfie Hunt List

1.     Illustrate geometric thinking

Not a Cat on a Hot Tin Roof, but a Tangent on the Beach
(courtesy of Emmanuelle Forgeoux)

2.     Illustrate the definition of a function

3.     Illustrate the notion of asymptote

4.     Linear Function Selfie

5.     Quadratic Function Selfie 

6.     Exponential Function Selfie

7.     Trigonometric Function Selfie

8.     Tangent line Selfie

9.     Concavity Selfie

10.  Illustrate Function Transformation

11.  Illustrate Inverse Function

12.  Illustrate Composition of functions

13. Illustrate Step wise functions

Upcoming activities which are still under development and will be described later in this blog include fitting an equation to a selfie using one of the commonly used geometry software, as well as organizing mathematical selfies exhibits for broader community involvement (think your school, the local library, coffee shops... etc).

Finally, other activities which we have not tried yet but were strongly suggested to us involve getting student mentors from older grades to work on mathematical selfies projects with the younger grades, therefore building a mathematical selfies community within the school, establishing communication and sharing of ideas. And last but not least, to build relevance to future uses of mathematics one can imagine tying a math selfies hunt to particular jobs that students have expressed an interest in, emphasizing the connection between mathematics and any potential future profession.

Reality and perception - by Kathy Jaqua

 Have you ever looked at a cloud and seen an old man’s face?  Did you squint your eyes to blur the distracting details so that you could see the image more clearly?  Did you really think there was an old man in that cloud? I’m guessing that you have seen a figure in a cloud, you did ignore the troublesome details that detracted from that image, and that you didn’t really believe what you saw was really there.  You moved between perception and reality seamlessly.

In theoretical mathematics, we need to be able to easily transition between what we know to be mathematically true, and how we can depict that knowledge.  Even drawing a graph of a simple linear function requires us to suspend the reality of a line being one-dimensional and infinitely long because we have to provide some width and some finite length for it to be visible and to fit on the page.  We generally aren’t picky about how thick the line is, and we add arrows to the end to indicate the infinite length; but in reality, we are only approximating the linear function. What we strive for is to capture the essence of the mathematical ideal through a “perfect enough” example that demonstrates that ideal.  This movement between mathematical reality and mathematical perception is inherent in discussions of mathematical knowledge and is a trait of math selfies. 

When students talk about their selfies, I often hear comments like “I know it’s not really like this, but it looks like it” to refer to some relationship being depicted.  Before reading my thoughts, look at the picture on the right and think about your mathematical perception of the trashcans and the reality of them. 

I think this selfie illustrates the distinction between reality and perception perfectly.  In this photo, the student noted the similarity of the trashcans, but went on to say that really they were all the same, but it looked like they were similar.  You might think this is a problem with selfies, but it actually revealed the student’s understanding of the difference between congruent and similar.  The student knew that all of the trashcans were congruent based on side lengths and angle congruence from the manufacturing process, while simultaneously noting that in the photo, perspective allowed the image to shrink uniformly creating an appearance of similarity.  The interweaving of the reality of congruence and the appearance of similarity was not a problem for this student or for me to understand his point. When showing this photo to others, the most common interpretation of what is being depicted is similarity.  What did you see?  Does the reality of congruent trashcans negate the perception of similarity shown in this photo?  I don’t think so.

Here’s another kind of distinction between reality and perception.  In the photo on the left, the student in the selfie was interested in the shapes of the giraffe’s spots, which she saw as polygonal.  It wasn’t critical to her that the sides be perfectly linear.  It was as if someone had drawn the polygons free-handed to show a variety of straight sided shapes. She saw the reality of a mathematical object in the appearance of a giraffe’s spots. The fact that she could look beyond the amazing experience of touching a giraffe in Kenya to see the shapes of the spots on the coat demonstrates the kind of mathematical awareness that I want to encourage.  

A note on linear functions - by Axelle Faughn

The following article just caught my eyes a few weeks ago, looks like we have some spontaneous uses for selfies by students for modeling with mathematics:
http://www.dailymail.co.uk/femail/article-3784114/Skills-dance-floor-Virginia-student-goes-viral-using-math-calculate-angle-dab-dance-move.html

One of my own students this semester used dabbing as a representation of a linear function for one of his selfies assignments. Not only did his submission provide me with insight into my students' world, but as the article shows, it can be taken further to perform more rigorous computations. In the article above I find it  particularly fascinating that although Anicca was able to measure all three sides of her triangle, she decided to use the tangent trigonometric ratio to calculate the angle of elevation of her dab. Linear functions are characterized by a constant rate of change: the slope. In Anicca's example, the tangent ratio represents the change in y (rise) over the change in x (run), which is precisely the slope of the associated linear function supporting the hypotenuse. Two students, two locations, same idea!

As for other representations of linear functions, here are some pictorial examples collected during recent mathematical selfies "treasure hunts" where I asked participants to illustrate these ideas:

On the left we have a good visual for connecting slope (the steps) to the linear function represented by the handrail, or its parallel line on the wall just above the steps. One should consider discussing with students what happens if steps are higher (steeper line) versus deeper, and how such decisions might be made when building a staircase. An online search provides standards for step building, but also plenty of examples or situations where standards have to be adjusted. And what might happen if the handrail and the staircase did not have the same slope (i.e. were not parallel)?













On the right, the student was concerned with the location of the line in a set of coordinates and found natural features to illustrate this concept. The height of the branch up the tree would represent the y-intercept, or initial value, of the tree branch linear function, in other words the "b" in the equation y=mx+b.




And for those of you with esthetics considerations, I thought the student who submitted the following picture did a great job at capturing the idea of embedding a quote into the picture to give it an edgy feel emphasized by the night time shot. Not only do selfies allow us to bring in more concrete and engaging representations into the classroom, they appeal to the more creative, artistically inclined students who will often come out of their way to come up with esthetically pleasing submissions.

Welcome! -- What is a Mathematical Selfie? - by Axelle Faughn

Welcome to Axelle and Kathy's Math Selfies blog! 

As a math teacher and teacher educator, I always look for new ways to make sure mathematics is perceived as relevant to the various people I interact with. In this blog I will explore the visualization of mathematical concepts through photography. The idea is to take a look at how mathematics can be popularized and how overall motivation for engaging in mathematical activities can be increased by the use of visual mathematics embedded in a real-world context. This will hopefully lead the path to greater mathematical awareness for all. 

So anyway, what is a Mathematical Selfie you are asking? 
Well, a “selfie” is commonly defined as a self-portrait photograph. Here I shall expand this definition to mathematical selfies as external representations of one’s mathematical perception of the world, in other words, a self-portrait of your mathematical world. 

In turn, mathematical awareness, or mathematical mindfulness, are defined as knowledge or perception of the presence of mathematics all around us. The antonym to which we will label as mathematical blindness, a severe and widespread condition that allows people to go about their daily lives without ever noticing how much mathematics surrounds them. No, I am not talking about meditating on mathematical concepts -- although that could be interesting as well -- rather to develop a certain habit of mind that allows you to notice where the mathematics is in our world. So... breathe in, breathe out... and notice the rate, the flow, the volume. Yup, that simple.

Just to give you a better idea of what I am talking about, here is an example of a mathematical selfie I took this past summer as I was visiting a friend in France. The bridge and its reflection in the water provide a nice illustration of function transformation (reflection across the horizontal axis), with the line of symmetry being the water surface:

That's all for today, but in later posts I will share my work with students, some of their findings are fascinating. Stay tuned!