As mathematics has been developed to solve problems faced by our cultures, many words that are being used in mathematics have been borrowed from our spoken languages. Some words have been specifically invented only for mathematical purposes. The mixing of these words can create confusion amongst students. As the vocabulary we use in the classroom has wide implications for the learning of students, it is worthwhile to reflect on the uncertainty that it can arise and how to deal with them effectively.

In mathematics, some English terms can cause confusion because the common usage of the word differs from the mathematical usage of the word. This can lead to misinterpretation and thereby difficulty in conceptual understanding. Let us consider some examples. The common understanding of the word in the English language is marked as E, and the mathematical meaning is marked as M.

Acute : (E) Sharp or severe in effect

            (M) Smaller than a right angle

Angle:  (E) A viewpoint

            (M) The difference of the directions between two straight lines

Even    (E) flat or smooth

            (M) Exactly divisible by 2

Factor  (E) An element contributing to a result or situation

            (M) A divisor

Function (E) The purpose for which something is designed

               (M) A relation between two sets with a condition on the domain

Power  (E) Capability of doing or accomplishing something

            (M) Exponent of an expression

Rational (E) Agreeable to reason

              (M) Expressible as a ratio of two integers

Root     (E) The underground part of a plant or a source of something

            (M) A value that satisfies a given equation

Similar (E) Having a likeness or resemblance

            (M) Having the same shape but not necessarily the same size

Table   (E) Furniture with a flat top

            (M) An arrangement in parallel columns to exhibit a set of observations

Volume (E) A large quantity of items or loudness of sound

              (M) The extent of space occupied by a solid

There are also some words such as axiom, coefficient, equation, ordinate, isosceles and transcendental that are not commonly used outside the mathematics classroom. This is what separates the language of mathematics from other languages. However, the principles and foundations of mathematics remain consistent no matter what language one speaks. This is why mathematics is considered a universal language as mathematics can be comprehended whatever natural language one speaks.

Let me give some examples how the students may get confused with terminology.

Consider the words ‘minus’ and ‘subtraction’. It must be stressed to the students that ‘minus’ (-) is the symbol used to indicate the operation of subtraction. The expression 5 – (-3) should be referred to as ‘From 5, subtract minus 3’. For convenience if it is expressed as ‘5 minus, minus 3’, the student may misunderstand it and consider it as ‘5 minus 3’, ignoring the second ‘minus’ as done with the ‘plus’ sign.

When it comes to algebra, a term that is commonly used is ‘cancel’ to denote ‘divide’. If the question is to simplify 12x/8, and if the students are told that they can cancel both numerator and denominator by 4 and obtain the answer 3x/2, they might get the wrong idea that they can always cancel similar terms. Therefore, if the question was (12x + 5)/8, they will use this idea to incorrectly write a solution as (3x + 5) / 2.

Another commonly used phrase is ‘cross multiply’ to denote transpose. To solve the equation (2x+3)/2 = (3x-2)/5, if they are told that it can be done through cross multiplication to obtain 5(2x+3) = 2(3x – 2), they may wonder what needs to be done to solve(2x+3)/2 = (3x-2)/5 + 1.

The terminology used in a question may also bring confusion amongst students. A good example can be a question such as “Show that a number divisible by 6 is even”.It is not surprising to find answers of the form “24 is divisible by 6 and it is even”. The student has misinterpreted the question which meant ‘all’ to mean ‘some’. Perhaps the question could have been worded as “Show that every number divisible by 6 is even”.

Consider a question to find the hypotenuse of a right-angled triangle when given the other two sides such as:

Here, if you consider the English meaning of the word ‘find’, you cannot find fault with what the student had done. But what is expected in mathematics is different. Perhaps the question could have been stated as “Find the value of x”.

Similarly, there can be confusion with the words ‘or’ and ‘and’. In the English language, ‘or’ means one or the other but not both. ‘Chicken or Fish’ means either chicken or fish. But in mathematics ‘or’ is inclusive by convention.

The answer to find the set representing A or B is given as A U B = {1,2,3,4,5,7,9}.

Similarly, in English ‘and’ is a conjunction used to connect words. In English, ‘Apples and Oranges’ would mean that all apples and oranges are considered. However, in mathematics ‘and’ means that both have to be true. In the example above A and B represents the set {1,3,5}.

Teachers should be aware of these situations and explain the differences using appropriate examples. Students hear and use words in their daily lives that take on different meanings to what they use in the mathematics classroom. We need to realize that it can bring confusion in the minds of the students and be prepared to help them to make their mathematical journey more successful. The ability to use proper words is essential for developing mathematical proficiency as students need to communicate mathematically effectively either verbally or in written format. As such, the introduction and clarification of related vocabulary of the lesson becomes a crucial activity for the teachers.

R N A de Silva

rnades@gmail.com

The author is an educational consultant at the Overseas School of Colombo and a Senior Examiner in Mathematics at the International Baccalaureate Organization.