Teachers always emphasize the importance of mastering algebra before graduating from high school. But what exactly is algebra? Is it really as crucial as people claim? And why do so many students find algebra difficult? Join us in this series on algebra in school through the perspective of Dr. Keith Devlin.
Dr. Keith Devlin is a professor at Stanford University with over 80 research papers and 30 published books on mathematics, some of which have won prestigious awards in the U.S., including the Pythagoras Prize, the Carl Sagan Award, and the Peano Prize.
Why is math difficult for many students? How can we make it easier?
Algebraic thinking: A way of reasoning that allows us to solve complex problems more quickly and accurately.
The secrets that help young children learn multiplication tables and perform multiplication with ease.
Part 1: The Difference Between Arithmetic and Algebra – An Overview of Algebra in School
This article focuses specifically on arithmetic and algebra as taught in schools, which differ significantly from arithmetic and algebra in professional mathematics.
Answering the initial questions may seem easier than solving a typical algebra problem in school, but surprisingly, very few people can provide good answers.
Let’s start by clarifying an important point: algebra is not merely “arithmetic with letters.” At its most fundamental level, arithmetic and algebra represent two distinct ways of reasoning about numbers.
Understanding Arithmetic
At its core, arithmetic involves performing basic operations—addition, subtraction, multiplication, and division—to compute numerical values. It is the oldest branch of mathematics, dating back to the Sumerian civilization around 10,000 years ago. The Sumerians had reached a level of societal complexity where money was used as a measure of wealth and as a medium for exchanging goods and services. Eventually, physical monetary tokens led to abstract imprints on clay tablets, which we now recognize as humanity’s first numerical symbols. Over time, these numerical symbols carried their own abstract meanings: numbers. In other words, numbers originally emerged as representations of currency, and arithmetic developed as a means to use money in trade.
It’s important to note that counting objects predates both numbers and arithmetic by thousands of years. Evidence suggests that humans started counting possessions—such as family members, livestock, crops, and belongings—at least 35,000 years ago. Archaeologists have discovered ancient bones with tally marks, which anthropologists interpret as early accounting records. These tallies did not represent abstract numbers; they directly corresponded to physical objects in the real world.
A Historical Perspective on Arithmetic
The ancient bone tally records were a form of documentation, and what they directly reflected were objects in the world rather than abstract numbers.
Initially, arithmetic was not performed using symbolic notation as we are taught today. The modern symbolic representation of arithmetic evolved over centuries. It began in India during the first half of the first millennium AD, was adopted by Arab merchants in the latter half of the same millennium, and only reached Europe in the 13th century. This is why arithmetic is often referred to as “Hindu-Arabic arithmetic” today.
Before the adoption of Hindu-Arabic numeral notation, merchants performed calculations using complex finger-counting systems or abacuses. Symbolic arithmetic only started replacing written-out calculations in textbooks around the 15th century.
Many people find learning arithmetic difficult, but with enough practice, most students manage to succeed—at least enough to pass exams. The reason arithmetic is relatively easy to learn is that its fundamental concepts naturally arise in everyday life: when we count, measure, shop, produce goods, use phones, go to the bank, or check our favorite sports team’s score. Numbers may be abstract—we can’t see, touch, hear, or smell the number three—but they are closely tied to tangible objects in our world.
How Algebra Differs from Arithmetic
However, with algebra, we take one step further away from the real world. The variables x and y that we work with in algebra represent numbers, but they are always generalized numbers rather than specific values. And the human brain is not naturally wired to think at this level of abstraction. Developing algebraic reasoning requires significant effort and training.
The key takeaway is that solving algebra problems is a way of thinking—one that differs from arithmetic thinking. The formulas and equations involving x and y are merely written representations of that thought process, much like sheet music is a written representation of a melody. Just as you can play an instrument without reading sheet music, you can perform algebra without using symbols. In fact, merchants and other professionals used algebra for 3,000 years before symbolic algebra was introduced by the French mathematician François Viète in the 16th century. Today, this early form of algebra is known as “rhetorical algebra” or “verbal algebra,” as opposed to the symbolic algebra that dominates modern education.
Key Differences Between Arithmetic and Algebra
We can summarize the distinction between arithmetic and school algebra in several ways:
- Algebra is a way of logical reasoning rather than simply thinking about numbers.
- In arithmetic, you compute with specific numbers. In algebra, you reason logically about numbers. Algebra focuses on relationships, not just calculations. For example, the equation a + b = c (where c is a constant) expresses a relationship between two unknown numbers, while 3 + 5 = 8 is a specific calculation.
- Arithmetic is quantitative reasoning with numbers, whereas algebra is qualitative reasoning about numbers.
- In arithmetic, you determine a numerical value by working with given numbers. In algebra, you use variables to represent unknown values and apply logical reasoning to determine their values.
A Practical Example
Consider the following two problems:
- “I am 14 years old, and my older brother is 4 years older than me. How old is he?”
- This can be solved immediately using arithmetic: x = 14 + 4.
- “My brother is 2 years older than me. My younger sister is 5 years younger than me. She is currently 12 years old. How old will my brother be in 3 years?”
- This problem requires more reasoning and can be solved in multiple ways. One method is to define the unknown values using variables and express their relationships in an equation. For example, if k is my current age, then k – 5 = 12, and my brother’s age in 3 years is (k + 2) + 3.
Final Thoughts
From these differences, we can conclude that algebra is not simply arithmetic with letters or symbols.
For example, the following task is arithmetic, not algebra: solving the quadratic equation ax² + bx + c = 0 by substituting specific values for a, b, and c into a formula.
However, deriving the quadratic formula from scratch is an algebraic task. Similarly, solving a quadratic equation using completing the square or factorization instead of using a formula is also algebra.
When students first encounter algebra, they naturally attempt to solve problems using arithmetic reasoning. This is understandable given the effort they have already put into mastering arithmetic, and this method works for very simple algebra problems.
Ironically, students who are strong in arithmetic may struggle more with algebra because, beyond basic examples, algebra requires abandoning arithmetic reasoning in favor of algebraic reasoning.
(To be continued…)
