A Forgotten Treasure Restored in Classical Catholic Education
In today’s digital age, few children encounter mathematics in the same way children did just 150 years ago. But in the 19th century, before calculators and screens, students in America learned arithmetic through a simple yet profound system called Ray’s Arithmetic. More than a collection of textbooks, Ray’s Arithmetic is a method, a philosophy, and a tool of formation. Today, the Classical Liberal Arts Academy is restoring this time-tested program in the context of authentic Catholic education, placing it within a full liberal arts curriculum that respects both reason and divine revelation.
To understand Ray’s Arithmetic and its importance today, we must explore its origins, structure, contents, teaching method, and the role it plays in forming young Catholic minds in the Classical Liberal Arts Academy.
The Origins of Ray’s Arithmetic
Ray’s Arithmetic was authored in the mid-19th century by Joseph Ray, a teacher and mathematician in Ohio. He published the first volume of his arithmetic series in 1834, and over time, developed a complete set of arithmetic and algebra texts for primary through advanced levels.
Joseph Ray believed that mathematics was not just a tool for commerce or industry, but a foundation of sound reasoning. He saw arithmetic as a way to train the mind in order, clarity, and precision. These are not only intellectual virtues, but, as St. Augustine teaches, are reflections of the divine order itself. Mathematics reveals the mind of the Creator who “has ordered all things by measure and number and weight” (Wisdom 11:20). In this way, arithmetic becomes not just a subject, but a spiritual discipline.
Ray’s Arithmetic was soon adopted in thousands of American schools. For decades, it was the most widely used math curriculum in the United States. However, like many traditional works, it fell out of use in the 20th century as educational theories shifted toward utilitarian goals and away from classical forms of study.
The Philosophy Behind Ray’s Arithmetic
What sets Ray’s Arithmetic apart is not only the content but the spirit with which it is taught. Ray’s approach is logical and incremental. Each lesson builds carefully on the one before. More importantly, it assumes that children are capable of reason, reflection, and mastery. It does not seek to entertain or distract. Rather, it demands attention and diligence from the student. As Aristotle writes in the Nicomachean Ethics, “We become just by doing just acts, temperate by doing temperate acts, brave by doing brave acts.” Likewise, we become rational by thinking rationally—and arithmetic is the perfect training ground.
Ray’s method is consistent with the classical and Catholic understanding of education as the formation of the whole person—not merely the transmission of skills or information. As the Catechism teaches, education is “a primordial right” and its purpose is to “form the human person in the pursuit of his ultimate end” (CCC 2223). Ray’s Arithmetic contributes to this by teaching not only how to compute, but how to think.
The Arrangement of the Ray’s Arithmetic Series
The Ray’s Arithmetic series is not a single textbook, but a complete course of study designed to guide students from their first encounter with numbers to advanced mathematical reasoning. This course spans the elementary through college levels and was designed with the same deliberate structure and clarity that marked the great classical systems of education. Each volume is meant to be studied thoroughly, with mastery of one leading directly to the next. This structure promotes mental development and aligns beautifully with the Catholic principle that education must be orderly and suited to the natural development of the student’s faculties (Catechism of the Catholic Church, 2223).
Below is a comprehensive list of the books in the Ray’s Arithmetic series, along with a description of each book’s purpose and appropriate audience:
1. Ray’s Primary Arithmetic
Intended for: Young beginners (typically ages 5–7).
Purpose: Introduces children to the basic concepts of arithmetic through real-life examples and oral exercises.
Topics Covered: Counting, addition, subtraction, simple multiplication and division, reading numbers, and using money.
This book uses familiar and concrete examples to gently lead children from intuition to understanding. For instance, problems involving apples, coins, or days of the week help ground numerical concepts in everyday experience. This echoes the teaching method of Our Lord, who often used common things like seeds and coins to reveal deeper truths.
2. Ray’s Intellectual Arithmetic
Intended for: Early elementary students (typically ages 7–10).
Purpose: Develops mental calculation and reasoning skills through oral exercises and story problems.
Topics Covered: Mental addition and subtraction, fractions, ratios, and proportional reasoning.
Unlike many modern math books, this one places a strong emphasis on mental discipline. Students are expected to solve problems entirely in their heads and explain their reasoning clearly. This method strengthens memory, logic, and the ability to articulate thought—skills needed not only in mathematics but in all subjects, especially in Catholic apologetics and rhetoric.
3. Ray’s Practical Arithmetic
Intended for: Middle-grade students (typically ages 10–13).
Purpose: Provides a thorough course in all the basic operations of arithmetic and their real-world applications.
Topics Covered: Long division, decimals, fractions, percentage, profit and loss, simple interest, ratios, averages, and measurements.
This book was designed for mastery of real-world arithmetic. The problems are drawn from practical situations, including business transactions, carpentry, farming, and timekeeping. It trains students to reason clearly about everyday decisions, cultivating the virtue of prudence. The structure also supports the Thomistic method of first understanding definitions and principles, then applying them to specific situations.
4. Ray’s Higher Arithmetic
Intended for: Upper-grade or high school students (typically ages 13–16).
Purpose: Advances the student’s understanding of arithmetic into abstract and theoretical areas.
Topics Covered: Powers and roots, progression (arithmetic and geometric), permutations, interest and annuities, and compound proportion.
This book elevates arithmetic to the level of a true liberal art, showing how number theory and advanced operations are grounded in reason and lead to deeper knowledge of the created world. By this point in the series, students are prepared to appreciate mathematics not just as a tool but as a reflection of divine order, echoing the words of Wisdom 11:20: “You have arranged all things by measure and number and weight.”
5. Ray’s Elementary Algebra
Intended for: Secondary students beginning the study of abstract mathematics (typically ages 13–15).
Purpose: Introduces the student to algebraic thinking using Ray’s hallmark step-by-step method, emphasizing clarity and mastery over speed.
Topics Covered: Simple equations, polynomials, factoring, proportions, basic exponents, and foundational word problems.
This first course in algebra helps students transition from concrete arithmetic to the more abstract reasoning that higher studies require. It builds habits of logical thought and structured analysis, necessary not only for mathematics but also for philosophy and theology. St. Augustine reminds us that some truths are not visible to the senses but are seen only by the mind. Algebra cultivates this interior vision and prepares students to explore truths that transcend the material.
6. Ray’s Intermediate Algebra
Intended for: Advanced high school or early college students (typically ages 15–18).
Purpose: Deepens the student’s understanding of algebra and prepares him for advanced mathematical study.
Topics Covered: Quadratic equations, radicals, exponents and powers, logarithms, systems of equations, and more complex word problems.
This course extends the student’s ability to think symbolically and solve increasingly complex problems. The practice of following a chain of logical deductions strengthens the intellect for engaging in philosophical and theological reasoning, especially when studying Scholastic texts like the Summa Theologica. The exercises in Ray’s Intermediate Algebra train the mind to perceive order in complexity, a valuable habit when examining the intricacies of both natural law and divine revelation.
7. Ray’s Geometry and Trigonometry
Intended for: Advanced high school students or early college students.
Purpose: Introduces Euclidean geometry and elementary trigonometry in logical sequence.
Topics Covered: Lines, angles, triangles, circles, area, proofs, trigonometric functions.
This work aligns perfectly with the study of Euclid’s Elements, forming part of the Quadrivium, the upper division of the classical liberal arts. Geometry teaches clarity of definition, precision of language, and visual-spatial reasoning. As St. Thomas Aquinas emphasized, geometry leads the student from the sensible world to contemplation of the unchanging forms that reflect divine reason.
The Content of Ray’s Arithmetic
Unlike modern math books, Ray’s Arithmetic focuses less on colorful illustrations and more on substance. Each lesson is composed of three main parts:
Definitions and Principles – The student learns precise definitions, such as what a number is, or the meaning of multiplication.
Examples and Explanations – Sample problems are worked through in simple language.
Exercises – A wide variety of exercises are provided, including word problems that require both comprehension and computation.
In Ray’s Practical Arithmetic, for example, the student may encounter a question like this:
“If 3 men can build a wall in 4 days, how long will it take 6 men to build the same wall, all working at the same rate?”
Such problems require not only computation but reasoning, estimation, and proportion. They teach the student how to apply mathematics to the real world, just as St. Thomas Aquinas encourages students to “apply what they have learned to good use.”
In addition to computation, Ray’s Arithmetic includes questions on:
Time and calendar calculations
Measurements of length, weight, volume, and area
Currency and interest
Proportions and ratios
Roots and powers
Business and trade applications
These practical topics ensure that mathematics is not an abstract game but a tool for responsible Christian stewardship. It prepares students for both the duties of life and the clarity of thought needed for higher studies.
he Method of Teaching Arithmetic in the Classical Tradition
One of the most striking features of Ray’s Arithmetic is its method of oral recitation and mental calculation. Long before calculators or even pencils were common, children were trained to work math problems in their heads and explain their reasoning aloud.
This method serves several purposes:
It develops memory and concentration.
It sharpens verbal expression, helping students articulate complex ideas clearly.
It encourages mastery, not guesswork or reliance on tools.
This is especially helpful for Catholic students, whose education should form both intellect and speech. As Scripture says, “Out of the abundance of the heart, the mouth speaks” (Matthew 12:34). A well-trained student should not only know the right answer but be able to explain how and why it is correct.
In the Classical Liberal Arts Academy, students using Ray’s Arithmetic study in this traditional manner. Lessons are studied slowly and repeatedly, often aloud. Tests are not merely about finding the right number, but explaining the thought process that led there. This mirrors the Socratic tradition, which values dialogue and reasoned argument. It also echoes the Scholastic method of St. Thomas, who always begins with definitions, proceeds with distinctions, and draws precise conclusions.
Why Ray’s Arithmetic Matters for Catholic Students Today
Many modern Catholic homeschool programs use math curricula developed by secular publishers. While the content may be acceptable, the method and spirit often reflect modern values: speed over depth, entertainment over effort, or utility over truth. In contrast, Ray’s Arithmetic helps restore mathematics as a liberal art—a subject worthy of study for its own sake, as a reflection of eternal truths.
St. Augustine wrote that number is among the “eternal and unchangeable truths” that reveal the nature of God. When a student studies arithmetic rightly, he draws closer to the order and wisdom of the Creator. This is not just useful; it is beautiful and holy.
Besides that, Ray’s Arithmetic teaches habits of mind that benefit every subject: clarity, perseverance, logic, and attention. These habits are especially important in the age of distraction. By using a method that rewards focus and diligence, Ray’s Arithmetic forms students for prayer, study, and life.
Ray’s Arithmetic and Moral Formation
Arithmetic is not a moral subject in itself, but the way it is taught and studied has profound moral implications. In the Parable of the Talents, Jesus praises the servant who manages his master’s wealth wisely (Matthew 25:14–30). Good stewardship requires the ability to count, measure, and plan.
Ray’s Arithmetic teaches students to use numbers rightly and responsibly. It also disciplines the will. Students must practice, review, correct mistakes, and strive for mastery. In this way, the study of arithmetic becomes an occasion for the growth of virtue.
Aristotle teaches that education is the cultivation of virtue through habituation. Ray’s Arithmetic, though a math book, offers daily opportunities to cultivate prudence, justice, temperance, and fortitude. It trains students to think carefully, to work honestly, to persist through difficulty, and to seek truth.
How the Classical Liberal Arts Academy Uses Ray’s Arithmetic Today
In the Classical Liberal Arts Academy, Ray’s Arithmetic is not just an old textbook. It is a living part of the Academy’s full liberal arts curriculum, integrated with studies in grammar, logic, rhetoric, philosophy, and theology. Students are guided step-by-step through the Ray’s Arithmetic series, with careful assessments and support.
Parents are not required to be experts. The Academy provides lesson plans, guidance, and tutorials so that students can work independently with accountability. The aim is not to race through grade levels but to achieve true mastery. A student may take months to finish one level—and that is perfectly fine. What matters is understanding, not speed.
This approach ensures that when students move on to Algebra, Geometry, and beyond, they are fully prepared. More importantly, they have learned to love the truth and to labor diligently for wisdom.
A Return to Sanity and Simplicity
Catholic education in our time faces many challenges: poor standards, moral confusion, and digital distractions. But the solution is not complicated. As Aesop tells us, “Slow and steady wins the race.” Returning to the simple, clear, and truthful methods of the past can restore order to the chaos.
Ray’s Arithmetic is one of those methods. It does not promise excitement or instant gratification. But it does offer truth, beauty, and formation. It is a treasure from our past that still speaks today.
The Classical Liberal Arts Academy invites Catholic families to recover this treasure—to place arithmetic back where it belongs, not as a tool of the world, but as a path to wisdom and virtue.
Conclusion: The Eternal Value of a True Education
At its core, Ray’s Arithmetic teaches more than math. It teaches how to think clearly, speak truthfully, and act justly. These are not only academic skills; they are human virtues and Christian duties.
St. Thomas Aquinas reminds us that “truth is the good of the intellect.” In learning arithmetic through a program like Ray’s, students encounter truth every day—in numbers, patterns, and problems. They learn that truth is not created, but discovered. And in discovering it, they draw closer to the One who is Truth itself.
In an age when education often forgets its purpose, Ray’s Arithmetic offers a refreshing return to order, reason, and holiness. In the hands of the Classical Liberal Arts Academy, this forgotten classic is once again helping young Catholics learn to think clearly and live virtuously.
Mr. William C. Michael, O.P.
Headmaster
Classical Liberal Arts Academy
References
Catechism of the Catholic Church. 2nd ed. Vatican: Libreria Editrice Vaticana, 2011. https://www.vatican.va/archive/ENG0015/_INDEX.HTM
The Holy Bible: New American Bible, Revised Edition. Washington, DC: United States Conference of Catholic Bishops, 2011. https://www.bible.usccb.org/
Aquinas, Thomas. Summa Theologica. Translated by the Fathers of the English Dominican Province. 1920. https://www.newadvent.org/summa/
Aristotle. Nicomachean Ethics. Translated by W.D. Ross. https://classics.mit.edu/Aristotle/nicomachaen.html
Augustine of Hippo. The Confessions. Translated by R.S. Pine-Coffin. Penguin Classics, 1961.
Ray, Joseph. Ray’s Arithmetic Series. Various editions reprinted by Mott Media. https://www.mottmedia.com/
