Students learn math today by walking up a long staircase of individual topics.
Third graders take their first step with learning to add and subtract within one thousand, for example. They take the next step into multiplication, then division, fractions, and so on. The goal of learning math is boiled down to reaching the next “step” so that by the time students reach high school, they have successfully climbed enough of the staircase to survive algebra. We hope.
This type of progression through the vast terrain of mathematics is oversimplified into a sequence of topics to be learned. It makes it more difficult for students to truly learn the concepts placed before them, affecting everything from assessment scores to their readiness to tackle a new topic. Realigning our instructional goals on mathematical “big ideas” can change that.
The problem with the typical journey through K–12 math
In “5 ways to help students develop a growth mindset in math class,” I explored how ELA and math are dissimilar. Though both disciplines depend on topics of study for context, math treats the topics themselves as the learning progression.
In ELA, we don’t ask students to read articles about the migration patterns of whales for the sole purpose of knowing more facts about whales; we ask them to read topical information so they can practice applying the deeper, foundational reading skills that apply to synthesizing information about any subject. But in math class, this is not the case. All too often, the point of the lesson is to learn how to do one specific thing or understand one specific topic. Students then interpret the purpose of mathematics simply as knowing how to add fractions, find a percent, and graph a parabola. No wonder students struggle to find a mathematical identity in a discipline focused on a different rote skill every other day.
By treating mathematics as a to-do list of topics to be covered, we rob students of opportunities to explore the beauty of an interconnected discipline. Mathematics is a rich tapestry of profound ideas, unique situations, clever applications, and stunning visualizations. But for one reason or another, that glorious image of what mathematics could be is condensed into discrete, unconnected, and often boring topics to be learned one at a time.
Standards alone aren’t enough
I don’t believe the over compartmentalization of mathematics is always done knowingly. The intention behind having a consistent academic standards framework across a state or nation is usually good, and setting up such a structure is no small feat. But the existence of academic standards is not enough to guarantee high-quality instruction or instructional materials for all. Plus, how we choose to play out those standards at a district, school, and classroom level reveals what we value about mathematics education.
Some classrooms treat the standards as the exact roadmap for instruction, using them verbatim to write student objectives on the board at the beginning of a lesson and to evaluate whether a teacher has done their job. Other classrooms literally march through the list of standards in written order from August to June. What starts as a general map for exploring the vast mathematical terrain of the year turns into exact turn-by-turn directions for how to cross (and then escape) that terrain as efficiently as possible.
Both approaches make us lose the forest for the trees. They mean we rarely, if ever, get past the surface level of mathematics. We go from topic to topic, explicitly teaching a procedure, practicing it at length, and assessing it before moving on to the next one. Each topic is disparate from others because they are taught and assessed separately from each other.
But what if? What if we refocused our goals, intentions, and philosophy of mathematics on the deep connections between topics? What if we spent time understanding how each new topic is not so isolated from others explored in the past? What if we came to realize that these deep connections and mathematical themes form a framework and progression all their own?
Mathematical truths are a better north star
Beneath the surface of our instruction lies a set of mathematical truths. A mathematical truth brings together understandings from a variety of topics into a centralized idea that is true in all mathematical domains.
Randall Charles named these truths “big ideas” and outlined 21 ideas for K–8 instruction in his 2005 article “Big ideas and understandings as the foundation for elementary and middle school mathematics.” These have been used as a lens through which to view the academic standards required by a state. Charles presents mathematical “big ideas” as “a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole,” further defining one of the original principles issued by the National Council for Teachers of Mathematics (NCTM) that “Teachers need to understand the big ideas of mathematics and be able to represent mathematics as a coherent and connected enterprise.”
Anne Watson, Keith Jones, and Dave Pratt call these truths “key ideas” in their book Key Ideas in Teaching Mathematics: Research-Based Guidance for Ages 9–19. They explain how several key ideas that transcend grade-level instruction are best implemented, and they provide notes about specific instructional approaches including introducing definitions, graphical representation, and purposeful experiences. They pay special attention to the art of developing conceptual understanding in students, especially before and throughout when abstract mathematics comes into play.
This deep, conceptual understanding matters cognitively, not just philosophically. Research into how students learn mathematics shows that conceptual understanding and procedural fluency support each other, so teaching toward only one or the other limits growth in both. Conceptual understanding in particular often takes a back seat to teaching a procedure when we are crunched for time. But by placing every instructional action we take into a framework of “big ideas,” we are able to fold in conceptual understanding at multiple points of learning, allowing conceptual understanding and procedural fluency to work together.
Further to the point, analysis of how humans learn in general indicates how crucial “big ideas” are, justifying that experts in a subject organize their knowledge around “big ideas.” This helps them build a strong foundation for the information they take in and supports their ability to transfer that knowledge to new domains.
The effects of shifting to the mathematical “big ideas” model
Shifting the purpose of mathematics to internalizing “big ideas,” as opposed to merely understanding individual topics, brings about immediate gains. Math teachers can see differences in how they view themselves; how they teach, talk about, and assess math; and how they design curriculum and systems.
How we view ourselves
When we prioritize a “big ideas” model in mathematics:
- We gain stronger horizontal and vertical alignment in the conceptual progression of mathematics across K–12. Because the learning progressions are rooted in connective themes, there is stronger alignment than with disparate topics. Research by the Consortium for Policy Research in Education (CPRE) also calls for progressions since the order of content explored is built not only on discipline logic and philosophy but also on the evidence of student thinking as well.
- Mathematics itself transforms from a set of skills you either do or don’t have and into a system of connected ideas, a system you can spend your life refining.
- Our identity shifts with a growth mindset. Our self-talk can transition from “I can’t do math because I’m terrible with fractions” to “I understand the fundamental ideas of equivalence and how they transfer to new mathematical situations.” Our engagement is based on feelings of competence or mastery and ultimately drives identity formation in the discipline. With extended engagement and confidence in math, we can find our own agency and adaptability too.
How we teach mathematics
A “big ideas” model of math instruction helps us:
- Leverage conceptual understanding naturally as a step to address prior knowledge and also bring lessons full circle, which can impact student learning most significantly in the long term.
- Connect a day’s learning across topics, domains, and even years of study. “Big ideas” help us look for connections deeper than surface level and identify how our teaching aligns with current understandings and builds toward what is yet to come.
- Prioritize teacher decision-making and framing of the math experience. We can choose and design tasks based on the ideas they reveal, not just the standard to which they align. We can invest more time in helping students recognize patterns, discuss theories, generalize, and justify.
How we talk about mathematics
The “big ideas” model of mathematics instruction can even change the way we speak about math:
- The language we use illuminates the true purpose of what we’re teaching to students, that is, it actually defines what counts as mathematical knowledge. We clearly show students “when they are ever going to use this in real life” by how we speak, because a connective idea focused on the importance of estimation, for example, is far more applicable to real-world scenarios than any individual mathematics topic.
- The focus in our classroom is built upon discourse and teachers posing the right kinds of questions to promote that discourse. We move from answer validation to thinking validation, with discourse as the mechanism by which that thinking is visible and sharable. Mathematics itself becomes less artificial because we reason through it together using connective ideas.
How we assess mathematics
Once we have let mathematical “big ideas” change how we view ourselves as mathematicians and how we teach and talk about the discipline, we can begin to analyze the information we get from assessment:
- Questions can become less about a single correct answer and more about explaining reasoning, comparing differing solutions, identifying errors, and making connections. This moves us from defining proficiency as “getting the right answer” to proficiency as “using the right kind of thinking.”
- We can measure the conceptual understanding that should be developing alongside procedural fluency, and we can also implicitly make known what we value as outcomes of learning. Students learn what math is based on how we assess it. When we fold in questions to get at the deeper truths of mathematics, students see success as thinking well, not getting lucky or remembering the answer.
How we design curriculum and systems
Finally, prioritizing a “big ideas” model in mathematics affects how we design curriculum and systems:
- Standards still play an integral role, not as the goal of instruction, but as instances of the “big ideas” in which they sit. Our learning progressions move away from rigid calendars of which topic to introduce and when, favoring the research-backed scope and sequence of mathematical concept formation.
- Correctly built conceptual understanding creates connections that support long-term retention and transferability. Therefore, summer slides are less drastic because “big ideas” are retained even if procedures fade. Students return to school in the fall with conceptual anchors that can be revitalized, rather than starting from scratch.
- Expectations shift. Though our current systems remain heavily constrained by demands for content coverage and testing pressure, reframing with mathematical “big ideas” could start chipping away at those expectations to build a learning progression worthy of and reflective of the twenty-first century student.
A new goal
Ultimately, the goal of mathematics education cannot be to simply move students one step higher on a staircase of disconnected skills. When we reduce math to a sequence of topics to cover, we limit both what students understand about the discipline and how they see themselves as mathematicians. A more productive shift requires us to start viewing mathematical topics and skills as the entry points rather than the endpoints. When we reframe mathematics around its “big ideas,” we invite students into something far richer: a coherent, connected discipline that makes sense, transfers across contexts, and grows with them over time.
As educators, the challenge before us is to design experiences that illuminate mathematical truths, prioritize right thinking over right answers, and measure success by the strength of students’ understanding rather than the number of topics they have “covered.”
