“Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution.” – Albert Einstein
The ultimate goal of the Five-Step Curricular Process is to guide students to a place in which their own, intuitive understanding of the world includes a fluency in the computational processes and symbolic representations of mathematics. Rather than have them commit a semester to rote memorization, the process allows the mathematics to be found inside their everyday lives. Beginning from a shared physical experience to modeling it to discussing it as a group allows them to negotiate the details of their ordinary experiences. By discussing the event’s features, they begin to conceptualize a more formal and specific way of communicating mathematical or logical concepts. Finally, the students are asked to represent what they have done symbolically.
Another major aspect of the Five-Step Curricular Process is to stimulate creativity, imagination, and student agency. It is important that the process allows students to view themselves as mathematicians, and that their own ideas and ways of conceptualizing math are important. After all, if one were to follow the infinite regress of proofs and axioms all the way down, the reason we do math the way we do is simply because it’s the best way anyone at the time could imagine.
This is important to note because while, in Algebra Project classrooms, the Symbolic Representation step does sometimes include traditional mathematical notation, it also sometimes doesn’t. It is incumbent on the students to determine what kind of notation makes sense to them and what they think will make sense to their peers.
Destini Chambers used musical notation.
Destini, currently a senior at Kennesaw State University where she will graduate with a Bachelor’s in Electrical Engineering dreams of working in the power infrastructure industry, using her robust skillset to negotiate the ever-changing utility-development landscape of budget restrictions and regulations. Even her hobbies represent her predisposition for computational thinking: Learning new languages and reading Agatha Christie’s Hercule Poirot series.
Like many students in STEM programs, she wasn’t bad at math. But her relationship with it wasn’t great either.
“It’s just something I have to do. I didn’t really give math much thought. It’s just something that I have to just push through, especially regarding my major. I just have to hunker down and get through this math stuff.”
When she was in her Sophomore year at Kennesaw, she received an email soliciting applicants for College Math Literacy Workers (CMLWs) for the Algebra Project. She applied, was accepted, and her first introduction to the Five-Step Process began that summer.
Of the Five-Step Process, she recalls, “At first I didn’t necessarily understand why we were doing it this way. The shared experience, the drawing, the group discussions. I wasn’t necessarily sure why we were doing these things. But when we got towards Feature Talk and Symbolic Representation, it tied it in for me to be like, ‘Oh, this is why we’re doing it.’ And it was really good just to learn why we were teaching like this, teaching these students not necessarily how to do problems per se or how to compute specific problems, but how to think around those problems and just to understand the whole process.”
During Destini’s summer learning the Five-Step Process, the CMLWs studied the Road Coloring Problem. The Road Coloring module involves a structured diagram of buildings, and roads leaving and entering each one, wherein students are tasked with developing a set of instructions for getting from building to building. For Destini, one of the most interesting aspects was being able to see how her peers conceptualized the problem.
“Seeing other people’s cities and designs helped me understand what the Road Coloring process is and why it’s important, specifically how different people’s brains connect to that process and how that can help with other math disciplines.”
She noticed how, as a CMLW tutoring 6th through 8th graders, she could see the conceptual shift in students when they were given latitude to determine their own symbolic representations.
“With that fifth step, I saw that they wanted to represent the math with what they know. So, students will be like, ‘Oh, this is the city, but it’s built with PS4s and Xboxes. This is a city, but it’s made with burgers and fries.’ It’s very individualized. They make the concept individualized to themselves.”
Destini found herself bringing her unique understanding of the world to the mathematical table. She had a long history with music, and to her, music wasn’t too far off from mathematics.
“So in high school, I had been in band for all four years. When I started college, I still played music with my guitar. And I was trying to play with the keyboard. I never took lessons in piano, but I took guitar lessons.
“Whenever we were asked to make this symbolic representation, I was trying to think of a way to make it different because the first thing I was thinking about was a visual representation, but then I realized that there were numbers involved. So, I was thinking about how a scale in music also has numbers if you count the number of notes within the scale. So then I thought maybe I could connect that to the symbolic representation and each building could be like a note or something. And whenever they’re leaving, I could just tie it into the scale system. That’s how I got to my symbolic representation.”
Destini imagined the Road Coloring problem as a musical composition played on a piano. Each building in a city corresponded to a note on a musical scale, specifically the C scale. In her city with four buildings, the first building represented the note C, the second building was D, the third was E, and the fourth was F.
Since sheet music for the piano has two sets of lines and symbols, one for the higher notes (treble clef) and one for the lower notes (bass clef), in her city design, the red roads represented transitions of notes in the treble clef, and the blue roads represented transitions in the bass clef.
Turning this concept into music, each measure corresponded to a building and its transitions. In her first city plan, all the notes played simultaneously because there was no specific sequence.
The box contains the written instructions for the road coloring sequence. For example, in Building 1, 1 moves to 4 via the red route, and 1 moves to 2 via the blue route, which corresponds to a transition from C to F on the treble clef and a transition from C to D on the bass clef.
However, when there is a sequence, like the pattern red-red-blue, the music is composed differently. It’s played in 3/4 time (due to the pattern red-red-blue being three transitions which are portrayed in a regular, four-beat measure. This also makes it a waltz).
Destini’s unique approach surprised everyone in the workshop, including the PD specialist. While she excelled in the Symbolic Representation step, it took a while to get there. As she previously noted, when she first began the Five-Step Process, she didn’t quite grasp what the point of it all was. Despite this, she said it wasn’t difficult to stay engaged.
“The discussions kept me engaged. Particularly just hearing other people’s opinions on what they were doing and what they thought as well, that kept me engaged through the process.”
That ability to empathize with one’s colleagues and peers is an important facet of the Five-Step Process. Being able to understand, or take the necessary time to understand, how another person conceptualizes a problem is as important as one’s own conceptualization. And Destini continues to use that as a tool in her day-to-day life.
“I think that whenever you’re doing specific problems or whenever you’re focusing on how the problem is done conceptually, it can help you to have this ability to view it from different perspectives. You can connect with others because maybe someone else doesn’t think like me, and someone else might need a wider view of how to do the problem or how to understand what’s going on. Not only just helping others with that understanding, but also taking it with you to help with other problems that you encounter, whether that be in work or in school.”
Destini doesn’t work as a Math Literacy Worker anymore, but she’s noted that her relationship with math has remained forever changed.
“It helps me build on other problems that I can encounter later that I wouldn’t have a computer to help me with. I know with things like ChatGPT, you can ask it all these things: What is this, what is that? But it would be to your detriment if you didn’t necessarily understand what you’re doing, be it with math or with engineering. You have to understand why you’re doing it. And I guess with the human conceptualization of math, just understanding why you’re doing it and having that toolbox to help you in the future.”
When asked what she would like to impart on students after all she has learned and been through, she is quick to reply, “Just how important imagination and creativity are to math. Most students think of math as either you just have to get it done or they hate math because of how they’re taught in school with how strict learning about math is. And if, starting from a young age, we just teach kids that math is fun, math can be fun, math doesn’t necessarily just have to be numbers, it could be burgers and fries, it could be Playstations and Xboxes, it could be music, math can be so many different things, and if you just learn that from a young age, I think your mind will shift on what math is. Math doesn’t just become math. It’s something deeper.”