An Introduction
If a friend were to tell you they never learned to read or write, you would be surprised and concerned. If a close friend admitted they never really understood math, how many of us would chirp, “oh, me, neither?”
The Algebra Project seeks to “raise the floor” of mathematics literacy in K-12 public schools by creating opportunities that increase access to and understanding of the language of mathematics, in collaboration with students, teachers, administrators, mathematicians, educators, researchers, and community activists. The project’s Five-Step Curricular Process is a pivotal tool in opening access to K-12 mathematics. Informed by lessons learned from the Mississippi theater of the Civil Rights Movement of the 1960s, it also incorporates theories and practices from progressive and experiential education, together with philosophies of mathematics.
The Algebra Project’s founder and late President Bob Moses once reflected on how to teach mathematics with a simple statement: “To do the math, you need students’ attention, and they need to be able to count.” The first proves much more difficult than the second as traditional approaches too often fail to capture student engagement. What is required is a transformative and culturally relevant curriculum that students and teachers readily engage in.
The Algebra Project believes that just as reading and writing literacy were the requisites for enjoying the full privileges of 20th-century citizenship, math literacy is added, along with reading and writing, as a key literacy for full participation in 21st-century citizenship.
Moses linked the literacy requirements of the 20th and 21st centuries when he said, “math and science now are front and center with reading and writing. If people don’t have these skills, then they really can’t participate as citizens just as people who couldn’t read and write and didn’t have the right to vote couldn’t participate as citizens back in the sixties.”
In their 2001 book Radical Equations – Civil Rights from Mississippi to the Algebra Project, Charlie Cobb and Bob Moses wrote, “almost anyone driving a car today is driving a wheeled computer…And as the need for assembly line workers diminished, the need for what economists have begun to call the “knowledge worker” grew.” Computers, the Internet, and the dot com bubble dominated the American culture of the 1990s and what some saw as a passing fad, Moses recognized as a new normal, not only for educational and economic access, but also for full participation in society and citizenship.
So, the question became for Moses, how do we transform the teaching of mathematics so that students become mathematically literate in both concrete and abstract arenas?
The answer became a combination of experiential and collaborative learning, all tied together with noted American logician Willard Van Orman Quine’s philosophy of regimented discourse.
Moses’ doctoral studies at Harvard, where he focused on philosophy before becoming a math teacher at The Horace Mann School in the late 1950s, centered around Quine. It was Quine who instilled in Moses a belief that math was structured in such a way that no human would ever naturally speak it. It, then, had its own grammatical and syntactical rules. This impeded the natural development of math literacy because, although the grammar and syntax of discursive language continued to constantly assert itself in people’s daily lives, the grammar and syntax of mathematics had to be sought out and learned.
Quine believed that mathematics began by the “regimentation of ordinary discourse, mathematization in situ.” Many of Quine’s teachings and beliefs became the cornerstone of what would later become Moses’ Five-Step Curricular Process, or a method in which one can teach both the syntax and semantics of math.
Utilizing what Moses referred to as ‘people talk’ or every-day, informal conversation, he found children were much more empowered to describe and tackle complex mathematical scenarios than when limited to only use the formulaic and computational language prescribed by traditional mathematics.
The Algebra Project begins instruction with a shared experience, such as a trip, that can be used by students as they begin explaining its features from this casual, conversational perspective before shifting to using symbols to operationalize those same features. This allows students to see both the abstract behavior of the mathematics, as well as being continuously grounded in real-world applicability.
While this Five-Step Process was created to be used in conjunction with Algebra Project curricula, it can and is also used in support of other curricula. The Algebra Project has primarily used this method in middle schools and high schools, but it has also been adapted for pre-K-5 settings, as well as college settings.
It is also important to note that when the Algebra Project works alongside teachers to teach the utility of the Five-Step Process, teachers are NOT asked to throw away their already established instructional practices. Rather, their expertise and knowledge needs to be brought forward and synthesized with these approaches. The Five-Step Process becomes a way of expanding the tools teachers bring to the teaching and learning of mathematics. The Algebra Project asks teachers to make some different moves in classroom discussions, such as prioritizing and acknowledging student ideas, looking at math from the perspectives of not only rules but also of meaning, and making students’ and teachers’ thought processes public and transparent. All of this classroom activity should be done building upon a teacher’s pre-existing skillset. This is a lot to ask of teachers, but it is what we are asking.
The Five-Step Curricular Process
The Algebra Project’s Five-Step Curricular Process is a method that brings math learners from a shared concrete experience to a solid understanding of a targeted mathematical concept and its representation in abstract mathematical symbols. Teachers themselves work through this process as a means of reconceptualizing and looking deeper into the targeted mathematical concepts before bringing this process to their students in the classroom. This series aims to provide a complex look into each step, why it works, and how to utilize it, as well as concrete examples of its use. In this intro, we will delve into each step with enough explanation and context to give you a working understanding of each one. The later installments will provide a deeper level of understanding so that you might use the process in your own life or teaching.
Step 1: A shared concrete event
Students come to the table equipped with extensive mathematical knowledge, both explicit and implicit. Much of that knowledge is just not yet translated into an explicit representation of mathematics. But through a shared event as simple as taking the subway, as many kids do to get to school every single day, students necessarily have a fundamental idea of a trip from which can be drawn basic mathematical concepts and relations that extend from the arithmetic of signed numbers to the beginnings of algebraic equations. The issue is explicitly excavating these concrete experiences and drawing out their mathematical implications. If students build the analogy between movement on a subway and translations on a number line, then they have a grounding for understanding the arithmetic of signed numbers which is a key concept of algebra. The trips are acting as a grounding metaphor that gives students access to what might otherwise involve numerical computations in the absence of understanding. Thus the Five-Step Curricular Process is meeting students in their real-world where their everyday knowledge is engaged as the first step. This connection between students’ prior knowledge and the more abstract mathematical formulations of concepts is a critical component of effective learning in the mathematics classroom.
Beginning with real-world events, like taking the subway, is crucial to shifting students’ mindset to mindful algebraic and computational thinking, but also in empowering the students with a sense of the value of the knowledge that they bring to mathematics. By taking a trip, and discussing the trip, there is a level of accessibility and participation such that no student feels left behind. Students may not yet understand how the experience relates to algebra, but what is important is their engagement, and not coming into the experience having already written off their ability to do the math. Instead, teachers create a classroom environment where students can affirm: “I can do this, I understand this, I do this all the time.”
Step 2: A picture/model
In the next step, students are asked to draw or model the event. In this stage, students focus on their visual representations of the event. This means that within the class, students are providing multiple perspectives on how one shared common event can be illustrated.
Students are given a wide breadth to picture their experience, such as the trip, in whatever manner makes sense to them. That may be a map, a picture, a graph, or anything else. Students are then asked to explain their pictures, first in small groups and then to the whole class. These pictures foreshadow representations that students will later make for certain mathematical ideas. In the process of sharing their pictures, students may be asked questions, and perhaps need to defend their intended meaning behind the pictures. It is important to allow students their self-expression in the process of gaining ownership over mathematical ideas and concepts before introducing the more conventional abstract representations of mathematics.
Step 3: People Talk
People Talk is the opportunity for students to discuss that shared concrete experience in their own words from their perspectives. This curricular process is where students first can put their voice on what will become the mathematical table.
In the realm of mathematics, by describing the event using their natural language and negotiating aspects of the shared experience in a conversational manner, students are developing their ownership and understanding of ideas which will ground the mathematics to be developed. Students’ voice becomes a critical driver in the growth of these mathematical concepts.
Step 4: Feature Talk
Mathematics is a conceptual language, but a language that no one speaks. Feature Talk is a construct to support students’ ability to give an interpretive reading of mathematics. At this stage in the process, two important tasks are set for students.
1. Feature identification. Students are presented with an essential or critical question concerning the shared event. First individually, and then through a discussion in their groups, students decide upon the features of the event which are most important in characterizing their response. That entails what we call feature identification.
2. Feature relations. Students are then asked to determine the relationship that they see among these features. That is known as feature relations.
It is both the set of features and the relationships among them that later will be used in capturing the mathematics in an iconic or symbolic representation.
Step 5: Symbolic Representation
In this final step, students construct symbols to represent the features they’ve identified. These symbols for the features are combined into a symbolic representation that captures the relationship among features that the students determined in the previous phase. These “symbol sentences” are constructed to do the same conceptual work as the conventional symbols of mathematics. By applying their symbolic representations to related problems, students have an opportunity to observe and understand through their symbols how the conventional symbols of mathematics work.
Within the classroom, teams of students share their symbolic representations so that each team understands how the symbols of other teams represent the targeted mathematical concepts and can be applied to additional tasks. Based on this shared understanding, the teacher acting as a representative of the mathematics community can then introduce the conventional symbols of mathematics as a means of communicating mathematical ideas, not just within the classroom but within the broader mathematics community.
Next Steps
The Five-Step Curricular Process is generally revisited through professional development sessions annually with the teachers the Algebra Project collaborates with, and by utilizing a cohort model, those teachers stay with the same class of students for multiple years. The point is that not only does the process take a long time to learn, but an even longer time to execute. So, while the purpose behind this introductory article was to provide a working understanding of the basics of each step, the following articles will go much more in-depth into each step. In the meantime, one can learn more about the history of the Five-Step Process, Bob Moses, and the Algebra Project in Charlie Cobb and Bob Moses’ book Radical Equations – Civil Rights from Mississippi to the Algebra Project. There is also a free upcoming virtual conference co-hosted by the Algebra Project and the We the People-Math Literacy for All Alliance which will have presenters and workshops surrounding various pedagogies, including the Five-Step Process, which you can find more info about here.
Read the next installment, “Step One – a Shared/Physical Experience“