The Five-Step Curricular Process: In Conclusion

by | Mar 29, 2024

Throughout this series, we’ve delved into conversations with educators, thinkers from various disciplines, and experts in mathematics and computer science. Our aim was to illuminate the multifaceted nature of the Five-Step Curricular Process, a methodology not confined to secondary school math classrooms alone. Instead, it serves as a profound paradigm shift, offering a fresh perspective on how we perceive the world around us. This process isn’t merely about teaching math; it’s a transformative lens through which we deliberately step back from established notions of categorization. With the active participation of our community, we engage in observing the world’s fundamental elements, creating models, fostering discussions, negotiating features, and collectively arriving at new insights. The significance of this process lies in its ability to foster collaborative learning and critical thinking, transcending disciplinary boundaries to cultivate a deeper understanding of the world and its complexities.

Student, Bob Moses, Lynne Godfrey. Feb 21st, 1993//New York Times Magazine

Lynne Godfrey, a long-time teacher in Boston and Cambridge, Massachusetts, began her career at a small private school on Beacon Hill in Boston. After a year she transferred to the King Open School, an alternative public school, where she taught 5th and 6th grade. Lynne describes her teacher training program as progressive except for one area.

“Math. We learned to teach math the same way I had been taught math. I do, you do, we do, where the teacher shows the class what to do, students apply the procedure or algorithm to a set of similar problems, and then the class reviews, clarifying and correcting any mistakes in applying the procedure.”

At the King Open, Lynne and her colleague assessed all of the 5th and 6th graders using textbook assessments, data from standardized tests, and information from prior teachers to place students into ability groups: high, middle, and low.

“I took the highest performing group and the lowest performing group. My colleague took the kids that were kind of in the middle. And every year, you could see that the groups broke down across race and family income. The kids who were in the highest performing group were mostly white kids and the kids in the lower performing group were mostly Black and brown kids whose families were generational, long-time residents who lived and worked in the neighborhood.”

Bob Moses happened to be embedded in King Open during this time developing the Algebra Project’s Transition Curriculum. This was early in the Algebra Project’s infancy, and the Five-Step Curricular process had yet to be exposed to the decades of classroom trials, refinements, and case studies it has driven since. Still, Bob encouraged Lynne to adopt the novel pedagogical technique with her class.

There was just one catch: No more ability groups.

Step 1: A Shared Experience

“For me, it changed what math class looked like. There weren’t these two separate tables of kids where the white kids sat at one table and the Black and brown kids at another. This called for a new, yet familiar structure, in the math class where all of the kids came to the table. No one group of students came knowing more than anybody else, except for maybe knowing more than me! So, that first activity for the Trip Line was an experience riding the T. The T was something all of the middle school students, 5th and 6th graders, knew something about. That was their mode of transportation.” In urban areas dependent on mass transit more than car-centric infrastructure, something like a subway or a bus system becomes a shared knowledge point for all community members, including children. This proved an effective first step in bringing the previously separated groups together because they shared a similar level of understanding of riding the T.

“That was the way they got to and from their after-school activities, how they went to visit other kids in the neighborhood, went to visit family and friends outside of Cambridge, or went shopping downtown. They knew it.”

Step 2: A Picture or Model

“We had a lovely art teacher, Pam, who worked with us to create a pictorial representation of that experience.”

Students doing Trip Line

The second step of the Five-Step Curricular Process is creating a pictorial representation of the shared experience. This allows students to engage other strengths and interests, while also allowing them to intuit aspects of the shared experience that were most important to them. A major part of the work in mathematics is identifying which features are important and representing them. This is the first opportunity for students to be introduced to that idea while creating something they design and can invest in.

“People would walk down the hallway and just be in awe of the things that kids were drawing on the mural from their trip. ‘Oh! There’s the escalator at Alewife!’ or ‘That’s the trolley at Park!” And so, when you think about who contributes, you begin to see that everybody brings something to the table.”

Working together on an art piece, step two of the process, allows students to collaborate on equal footing and begins to dismantle status or hierarchies, and sets the stage for the social construction of math and the power of collective work in the math classroom.

Step 3: People Talk

People Talk may be the most well-known step of the Five-Step Curricular Process. Simply put, it is the practice of having a dialogue in ordinary language. Students write stories and talk about their experiences in their own, intuitive language. Each student’s experience is valid and valued. People Talk intentionally makes discussion accessible to all participants.

“Everybody showed up smart because everybody knew something. So, it wasn’t just an elite group who could,

you know, break the code, but everybody could contribute.”

When everyone’s voice in a classroom community is given equal importance, it doesn’t just empower those who are often used to staying quiet, it shifts the entire culture of dialogue among everyone. Step Three empowers students to integrate their own observations and cultural perspectives into the analysis of shared physical events. In Algebra Project classrooms, the notion of “culturally relevant” mathematics isn’t predetermined; rather, it emerges from the unique contributions of the students themselves.

“If you’re used to always being the person with the ideas, the person who’s always seen as, or told, you’re the smart one, you begin to see that if you listen that there are these other ideas that are out there. Ideas you might not at first think of as worth listening to. Holding that space for kids to do that creates a shift.”

Step 4: Feature Talk

“You’ve made some pictures or models of that experience. You’ve written about it. You’ve talked about it using your own words. And now we’re going to take from your language the features that get mathematized. That was like turning everything on its head. Instead of opening the textbook and being asked to read symbols and use algorithms we were starting with a physical experience, language we knew and understood, and representations we created. The textbooks started with the symbolic representations, whereas the Five-Step Process started from a shared experience.”

Just as Step Two allowed students to intuitively identify important features of an event, and People Talk allowed them to negotiate those ideas among themselves, Step Four makes those features explicit and focuses heavily on naming them something specific and prescriptive. The main goal of this step is to continue to allow conversation, not yet stepping into the traditional symbolic representation of mathematics, but reducing ambiguity while agreeing on a shared set of terms, meanings, and symbols.

“Building on a shared understanding and agreement about equivalent trips helped many students form a clearer concept of integers. They could think about integers as ‘how many and which way’. They discussed the concept of equivalence in many contexts in everyday life and in other math contexts like fractions, signs in the Chinese Zodiac, and algebraic expressions. Turns out equivalence is a really big idea in math!”

Step 5: Symbolic Representation

The last step of the Five-Step Curricular process ends where, as Lynne previously noted, most textbooks begin: symbolic representation. Once a shared experience has been discussed and modeled such that its features can be identified using shared language, you can take the abstract symbol system of mathematics, and formulate expressions around it.

“There’s something about taking something that you learned 30 years ago in high school and thinking about what it means in this context. It’s a fun way to learn algebra. I always liked algebra, the symbol manipulation, the properties, and formulas, but through the tasks in the Algebra Project Transition Curriculum I was developing a deeper conceptual understanding.”

By taking the time to honor and engage in the first four steps, Lynne and her students had a more enduring understanding of the meaning behind the abstract mathematics. When her class was split into ability groups in previous years, those at the top were simply more adept at memorizing and applying procedures. Now, every student had greater fluency with procedural knowledge and could reconstruct the meaning behind the symbols.

Conclusion

The Five-Step Curricular Process isn’t merely an instructional technique for mathematics; it’s a gateway to a new worldview. This approach cultivates collaborative exploration and critical inquiry. It’s not just about teaching facts and formulas; it’s about collectively reimagining how we perceive and interact with the world around us. Through this process, we transcend disciplinary boundaries, fostering a deeper understanding of our reality and unlocking pathways to innovative thinking and problem-solving.

For Greg Budzban speaking about the First Step, “When you’re literate in an area, it means that you could pick up something and you can engage with it. It’s accessible to you. I say that I can read because I can pick up a book and I can actually make

sense of it. So I think it’s learning a collection of ideas and perhaps more importantly, that we get students to the place where they see themselves as learners of mathematics.”

Our own conclusion on Step Two was that, “Mathematics is a human activity, and the classroom is a place – a meeting tool – where teachers and students develop common interests that coincide as an interest in mathematics. They do this when they work in small groups to negotiate ideas between each other.”

Alan Shaw, discussing Step Three, said, “By the time they’re in 12th grade, you ask a question and nobody raises their hand. Everybody’s afraid to speak, they don’t wanna be wrong. They don’t want somebody to tell them that something they said or thought wasn’t valid. And so teachers, early on, gotta work against that. And People Talk is one clear approach on how to work against that.”

Madeline Muntersbjorn, and Step Four, “Feature Talk is more precise and rigid. It’s very wordy. And this then makes it more natural for students to embrace abbreviations. So for me, where a lot of logic textbooks will say, ‘Here’s the symbols, put it in natural English,’ I say, here’s some natural English. We must put the natural language into an artificial language before it can be put into symbols.”

Destini Chambers on Step Five, “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.”

Since those early years at King Open, Lynne has continued working as a professional development specialist with the Algebra Project, working with other teachers to use the Five-Step Curricular Process.

“There can be pushback.” The most common issue for many teachers is time. They are concerned that if they take time for the 5-step process, especially the People Talk or discourse, they will not be able to ‘cover’ everything in their pacing guide.

“And they are probably right. You won’t be able to ‘cover’ everything, but their students will have a deeper conceptual understanding and will have learned a way of learning that can be applied to other, even sometimes unfamiliar math content.

If you can get people to stay with you long enough to work through the steps until they begin to see the formal representations of everything we’ve been talking about, that’s when they’re like ‘Oh my goodness, I didn’t know that was a way we could think through these things. We were talking about dividing fractions, but I wasn’t aware that that’s what we were talking about. I really understand it now, and that’s really powerful.’”

This leads Lynne to one conclusion.

“I think the same people who are teaching math now, just like the way I was taught to teach math, need a new vision of what it means to teach and what it means to learn. It’s like a light bulb turns on, the realization that you, the teacher, no longer need to know and do everything for the students. This realization can lead to a deeper understanding of math for students and teachers.”

Share This

Share this post with your friends!