written by: Dr. Tony McCaffrey
Students of mine who have great difficulty with algebra have little problem solving the puzzle below, which uses pictures of food instead of standard variables (i.e., x, y, and z). (The solution is at the end of this blog.) Further, two of my current summer students are a 3rd grader and a 4th grader and they can also solve the food puzzle below, even though they are at least two years away from studying algebraic equations in school.
Solving the food puzzle is equivalent to solving the system of equations below, which uses the variables x, y, and z instead of food pictures. These same students who struggle with algebra and can solve the food puzzle cannot solve the equivalent system of equations below. What is happening?
First, there are some students who can solve both versions and see the connection between the food pictures in one and the variables x, y, and z in the other. However, for those who can only solve the food picture version, they say that the food pictures “make sense to them” while the variables do not. The food pictures speak to them while the variables remain silent.
The food pictures (i.e., emojis) resemble what the problem is about: food items. The variables x, y, and z, however, are detached from any concrete situation. They could apply to just about any situation. This level of abstraction gives algebra its extreme power because it can describe relationships between quantities regardless of what scenario they come from. However, this same level of abstraction definitely has a downside. It makes it very difficult for some students to understand algebra. These students need things to be a bit more concrete and the food pictures, in particular, and emojis, in general, do just that. After all, mathematics is all about training the mind to see patterns and if emojis help a sizable group of students see and understand the underlying patterns better than standard variables do, then let’s use the notation that makes them better at detecting and comprehending patterns.
Second, the food puzzle presents a system of equations between three quantities related to tacos, burritos, and hot peppers. Three-variable systems of equations are not covered in Algebra I. Since all of my students have completed at most Algebra I (and some much less), none of my students has worked with a three-variable system of equations before. Yet somehow, the food picture version is intuitive enough for them to understand even though it presents an algebra topic they have never covered.
The American philosopher, C. S. Peirce, categorized signs in multifaceted ways and created the field of semiotics (Peirce 1955). In Peirce’s terms, emojis (e.g., food pictures) are icons because they have a resemblance to what they represent. The emoji taco resembles a taco. Standard algebra variables (e.g., x and y) are what Peirce would call symbols because they have an arbitrary connection to what they represent. In our example, someone proclaimed that x should be about tacos. For example, very often algebra teachers and textbooks will say something like, “Let x be the cost of a taco.” The symbol x does not look like a taco in any way. Instead, someone just declared that x will stand for a quantity related to tacos.
It has been well established in psychology research beginning as early as 1967 that icons, which resemble what they are representing, are easier to remember than textual symbols (i.e., individual letters as well as words), which have a haphazard connection to what they stand for (Shepherd 1967; Hockley 2008). With this significant memory advantage for icons, which is called the “picture superiority effect,” why don’t we use icons in algebra class instead of standard variables? Further, if three-variable systems of equations using emojis (i.e., icons) can be solved by students who have never studied such systems, then why don’t we use emojis in algebra class? In sum, emojis are easier to remember, more concrete, more intuitive, and easier to understand than textual symbols. So, let’s replace standard variables with emojis in algebra class and just imagine the consequences!
First, imagine the artistic fun we could now have in algebra class! Finding the best emojis to represent various quantities could become an art contest. Students would individualize their equations with their carefully selected emojis, thus giving the equations much more meaning and personal connection. For example, all three equations below express the same mathematical relationship: If a salesperson makes 2,000 dollars on each car sold, the equation expresses their profit by multiplying 2,000 by the number of cars sold by that salesperson. Students could debate about which equation presents the situation best artistically. Which equation is the most intuitive to understand given the original scenario? And, of course, which equation is mathematically correct? In this way, algebra is no longer just about being mathematically correct in describing the relationships between changing quantities. It is also about communicating how the equations best express the situations they are representing. Emojis help add a whole new dimension to algebra class. Now, algebra class can also be concerned with the art of communicating mathematical results by making math equations more intuitive and comprehensible, as well as finding the equations that accurately describe phenomena in the world.
Second, students who are talented sketchers will now be in high demand in algebra class to draw emojis on the papers of their classmates. Third, when students get tired of drawing emojis on their algebra homework, then perhaps they will look for a shorthand and perhaps want to use individual letters—thus, recreating for themselves the standard variables (i.e., x, y, and z). But now, the standard variables will be much more meaningful for the students because they saw a need for an abbreviated notation and created it themselves! Finally, teachers will already have one step in place when trying to transform algebra from a STEM subject (i.e., Science, Technology, Engineering, and Math) to a STEAM subject (i.e., which integrates Art into the subject).
Of course, we also teach our students to use the standard variables (i.e., x, y, and z) so that they can perform well on standardized tests and do well in college math classes that do not use emojis in their algebraic equations. Until the rest of the math world becomes enlightened enough to use emojis as variables, we will have to make accommodations for them and take into account their standard variables (i.e., x, y, and z), which for many students are more difficult to remember, intuit, understand, and apply to situations in the real world.
Solution to Puzzle:
Taco = 20
Burrito = 5
Hot Pepper = 1
Taco + Burrito + Hot Pepper = 26
References:
Hockley, W. E. (2008). The picture superiority effect in associative recognition. Memory & Cognition, 36(7), 1351-59.
Peirce, C. S. (1955). Philosophical Writings of Peirce. J. Buchler (Ed.). Mineola, NY: Dover Publications.
Shepard, R. N. (1967). Recognition memory for words, sentences, and pictures. Journal of Learning and Verbal Behavior, 6, 156-63.
Acknowledgment:
Nick Mandell, a summer student of mine, originally showed me the Food Puzzle problem, which led to many observations on my part as I gave it to all my students. Thank you, Nick, for making Emoji Algebra possible!