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Chess knights greet entrants seeking mathematical knowledge

Introduction: The Turkey Day Tirade –

During a recent Thanksgiving Day dinner, Justin, a master-level hydrologist and entrepreneur who operates a landscaping business specializing in exotic water features, voiced his frustration with an experienced employee (we’ll call him Joe) who was constantly asking for detailed instructions about what he was to do next. Justin wanted his employee to be able to assess a new (but familiar) problem and then figure out for himself what needed to be done. What Joe seemed to lack was a conceptual understanding of his job (and possibly the seriousness of the expectations of his employer).

This article contrasts conceptual and procedural learning. It explains how Chess in Education (CIE) reinforces conceptual learning, focusing specifically on the teaching of math.  CIE  serves as an efficient and effective tool to teachers in preparing students like Joe for 21st Century Skills.

For some highly regimented jobs, procedural learning is all that is needed. A fast food line cook doesn’t need the conceptual training of an executive chef. But the job market rewards conceptual thinking; jobs that are purely procedural are candidates for automation.

For educators trying to prepare students for productive careers, this is a familiar complaint from employers: new job applicants are often unprepared to meet the demands for creativity and critical thinking necessary in today’s job market.

What is Conceptual Versus Procedural Understanding?

“Conceptual understanding in math is the creation of a robust framework representing the numerous and interwoven relationships between mathematical ideas, patterns, and procedures. This framework can be used to coherently integrate new knowledge and solve unfamiliar problems.” Ki Karou, Director of ST Math Content

Math teachers versed in modern training pedagogy are familiar with the distinctions between conceptual knowledge versus procedural knowledge. Over the past decades, the conceptual approach has become accepted as best practice.

“From a neuroscience perspective, conceptual learning requires schemas*. Schemas are all about connections, and building schemas of mathematical concepts gives students the ability to solve problems they haven’t seen before.” Ki Karou, Director of ST Math Content

* Schemas: In psychology and cognitive science, a schema describes a pattern of thought or behavior that organizes categories of information and the relationships among them. Wikipedia

Students need to know not just the steps (i.e. procedures) to solve specific math problems, but to understand math as a robust framework representing numerous interwoven relationships among mathematical ideas, patterns, and procedures. Math is more than a textbook page of equations and formulas to be solved; math is about connections, visualization, the dynamics of change.

An easy way to understand the distinction between a conceptual and procedural approach is to think about getting travel directions. A procedural approach would lay out a series of directions and turns to make at each step of the journey from the start to the destination. If the directions are followed (and accurate), it would get the driver to their immediate destination. A conceptual approach would create a map to aid in getting to many future destinations.

Conceptual and procedural approaches each have their own merits, and often are most powerful when combined. GPS systems represent an integrated package of conceptual and procedural information that makes navigation easy.

Chess as a Tool in Teaching Math

Chess offers a powerful tool to build a conceptual understanding of math in children. While the Internet is awash in clever programs that gamify the teaching of early math, chess provides an immediate, direct, and tactile offline tool for teachers. With Chess in Education (CIE) training, chess becomes a simple, flexible tool that engages students and allows them to build a schema that connect abstract concepts, visual patterns, and written procedures. [See Resources below for examples.]

The Challenges of Bringing Conceptual Learning to the Classroom

According to Ki Karou, the conceptual learning approach changes the role of the Educator. Rather than building a pathway for imparting knowledge, a new aspect of the teacher’s  job is to create a mind map and help the student to explore how to use that map to solve problems, not just in the classroom but in life. This is a paradigm shift from the historical role of the teacher, but is very much in line with the rationale for Chess in Education.

Check out this half-hour podcast from the Mind Research Institute’s interview with Ki Karou for a deep dive into how the brain learns and what kinds of instruction can facilitate schema building for conceptual understanding.

Productive Failure

Karou also discusses the importance of “productive failure” as an aspect of conceptual learning.

Such failures are key to developing the resilience necessary to take on difficult challenges. In chess, Elliott Neff, founder of Chess4Life, describes the three possible outcomes to a game of chess as win, draw, or … learn. The role of a good chess instructor is not simply to critique the mistakes made in a single game, but to train the student in the conceptual process of analyzing his or her own games. On an emotional level, the teacher’s job is often to redirect the child’s disappointment in losing to understanding it as an opportunity for improvement and further discovery. This isn’t just a tactic to assuage hurt feelings. It is based upon hard research that shows the importance of resilience –of experiencing failure and overcoming it.


Conceptual learning is about building a robust schema in a student’s head.

The question for educators is, “How do I create an environment that is going to maximize exploration and discovery of all these different pathways and maps in order to build this larger schema?”

A secondary consideration is how to do this in large classrooms where there are already heavy demands on teachers. The answers lie in having tools that are efficient, effective, inexpensive and fun.

CIE checks all these boxes.


Chess and Mathematics in Primary Schools – a European Union program: ERASMUS.

Teaching Mathematics through Chess [ECU102] – a class offering by Chess Plus, a CIE Coalition member.

50 Chess and Mathematics Exercises for Schools – a sampling of Exercises from Chess Plus.

Chess in Schools – a US-based non-profit organization specializing in training, certification, and consulting to educators in the discipline of Chess in Education (CIE).

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