{"id":26797,"date":"2026-07-07T05:00:00","date_gmt":"2026-07-07T12:00:00","guid":{"rendered":"https:\/\/www.nwea.org\/blog\/?p=26797"},"modified":"2026-07-06T09:51:03","modified_gmt":"2026-07-06T16:51:03","slug":"mathematical-big-ideas-can-change-the-way-you-teach-and-reach-more-students","status":"publish","type":"post","link":"https:\/\/www.nwea.org\/blog\/2026\/mathematical-big-ideas-can-change-the-way-you-teach-and-reach-more-students\/","title":{"rendered":"Mathematical \u201cbig ideas\u201d can change the way you teach\u2014and reach more students"},"content":{"rendered":"

\"\"Students learn math today by walking up a long staircase of individual topics.<\/p>\n

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 \u201cstep\u201d so that by the time students reach high school, they have successfully climbed enough of the staircase to survive algebra. We hope.<\/p>\n

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 \u201cbig ideas\u201d can change that.<\/p>\n

The problem with the typical journey through K\u201312 math<\/h2>\n

In \u201c5 ways to help students develop a growth mindset in math class,\u201d<\/a> 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.<\/p>\n

In ELA, we don\u2019t 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<\/em> 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<\/a> in a discipline focused on a different rote skill every other day.<\/p>\n

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.<\/p>\n

Standards alone aren\u2019t enough<\/h2>\n

I don\u2019t 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<\/a> 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.<\/p>\n

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.<\/p>\n

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.<\/p>\n

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?<\/p>\n

Mathematical truths are a better north star<\/h2>\n

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.<\/p>\n

Randall Charles named these truths \u201cbig ideas\u201d and outlined 21 ideas for K\u20138 instruction in his 2005 article \u201cBig ideas and understandings as the foundation for elementary and middle school mathematics.\u201d<\/a> These have been used as a lens through which to view the academic standards required by a state. Charles presents mathematical \u201cbig ideas\u201d as \u201ca statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole,\u201d further defining one of the original principles issued by the National Council for Teachers of Mathematics (NCTM)<\/a> that \u201cTeachers need to understand the big ideas of mathematics and be able to represent mathematics as a coherent and connected enterprise.\u201d<\/p>\n

Anne Watson, Keith Jones, and Dave Pratt call these truths \u201ckey ideas\u201d in their book Key Ideas in Teaching Mathematics: Research-Based Guidance for Ages 9\u201319<\/em><\/a>. 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.<\/p>\n

This deep, conceptual understanding matters cognitively, not just philosophically. Research into how students learn mathematics<\/a> 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 \u201cbig ideas,\u201d we are able to fold in conceptual understanding at multiple points of learning, allowing conceptual understanding and procedural fluency to work together.<\/p>\n

Further to the point, analysis of how humans learn in general<\/a> indicates how crucial \u201cbig ideas\u201d are, justifying that experts in a subject organize their knowledge around \u201cbig ideas.\u201d This helps them build a strong foundation for the information they take in and supports their ability to transfer that knowledge to new domains.<\/p>\n

The effects of shifting to the mathematical \u201cbig ideas\u201d model<\/h2>\n

Shifting the purpose of mathematics to internalizing \u201cbig ideas,\u201d 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.<\/p>\n

How we view ourselves<\/em><\/h2>\n

When we prioritize a \u201cbig ideas\u201d model in mathematics:<\/p>\n