# Math History Can Help Students Overcome Learning Difficulties

## Negative numbers and mathematical notation are better understood by learners when they examine their historical origins.

By Donna Iadipaolo

In his 1997 article the “History of Mathematics Can Help Improve Instruction and Learning” Shmuel Avital states that math teachers should “examine history to understand learning difficulties.” Although Avital’s article was published fourteen years ago, his rationale and arguments are valid today and can be helpful for all math teachers.

**Negative Number Nemesis**

Many learners have difficulty with negative numbers. In my experience, they often do not immediately realize that a subtraction problem is the same as adding a negative number. For example, 6 - 4 is equivalent to 6 + (-4). They frequently wonder how a negative number multiplied by another negative number can possibly equal a positive number. Even the concept of what a negative number is can be difficult for some people to grasp. In fact, I have encountered graduate students who find it difficult to create a list of real-world examples using negative numbers. When asked, the only examples they come up with are financial debt, sea level, and temperature.

### Using History as a Way to Understand Math

Civilizations, and even brilliant mathematicians, have grappled with negative numbers throughout history. This will let them know that they are in good company. None of the rules of operations for negative numbers, for instance, have been found in the work of ancient Arab mathematicians, even though we know that they were aware that the Hindus formalized these rules. Additionally, it has been stated that the formidable mathematician Blaise Pascal (1623-1662) said he “saw no need for the introduction of negative numbers.” As late as the 19th century, English mathematician Augustus de Morgan (1806-1871) said he “considered numbers less than zero inconceivable.” As Avital states: “It is only in the nineteenth century that the negative numbers achieved a fully recognized position in the number system.” Furthermore, Avital points out that a study conducted by the Weizmann Institute of Science suggests that “exposing teachers to the historical development of negative numbers can help them improve their understanding of students’ difficulties in school."

**Grappling with “X”**

Negative numbers are not the only challenging math concept, many young mathematicians also become anxiety-ridden when they encounter variables. The fact that we have so many terms for "x" such as "some quantity" or "an unknown number" is emblematic of the ambiguity. Students also grapple with a number next to a variable implying multiplication, i.e. 2x+3x=2*x+3*y. Mathematics is full of symbols and notations which one must keep straight. It is almost, at times, like a foreign language or secret code. It is easy for people to get the meanings mixed up, especially when there are so many similar ways of writing an expression.

### Understanding Mathematical Symbols and Notation

It might encourage your learners to know that mathematical symbols and notation have evolved over time, and there were often conflicting methods. The Greek mathematician Diophantus used shortened words for operations. Babylonians and Sumarians used pictures to represent unknowns. The famous Arabic mathematician Al-Khwarizmi did not use any mathematical symbols whatsoever in his book *Hisab Al-jabr W'al Muqabala*, from which the term algebra was derived. The French mathematician Francois Viete (1540-1603) suggested the use of consonants for constants and vowels for variables. In short, when teachers understand how symbols and notations changed throughout history, they are more apt to have patience with pupils while they are getting the hang of symbols and notation. and the students, themselves, are more likely to approach their learnng process with more patience. Here are related lessons that may help assist with math learning challenges.

## Math Lessons and Activities:

Pupils create a math vocabulary puzzle using words from a math glossary. This math resource may be applied to review a variety of mathematical terms and definitions that learners are having trouble remembering.

Measuring the Earth (Eratosthenes' method)

Learners reproduce Eratosthenes’ measurement of the Earl. They use a geometric model to engage in problem solving, reasoning, communicating, representing, and solving challenges.

Applying Properties to Variables

Have some fun combining like terms. First, pupils use certain terms to name items (such as stars, moons, hearts) and then they use variables. They also apply the distributive property and write symbolic equations based on story problems.

Individually, or as a class, learners prepare budgets based on set totals. They then compare estimate costs versus real costs.