Mind your language in Mathematics
View(s):
For a long period of time in my educational career, I was working in the United World College of Hong Kong, where pre-university education and preparation is offered to selected students from all around the world. About 250 students usually representing over 80 countries follow the International Baccalaureate program annually in this institution. About half the number of students is selected through scholarship schemes granted by the National Committees of the United World Colleges movement, in their respective countries. Once every two years, a scholarship was granted to a deserving Sri Lankan student too.
I have had a close acquaintance with the Sri Lankan students, and I recognized their challenges in matters involving the change of the medium of instruction from their mother tongue to English. Being gifted students having been chosen through a highly selective process they usually manage to be in a comfortable footing within about a period of six months. While discussing about the difficulties on the following of various subjects with a new medium of instruction, I was surprised to find many of them stating that mathematics caused more problems than other subjects. This does not seem to make sense as one would have expected mathematics to be the easiest subject to follow as it has its own symbolic language. Why then has this situation arisen?
My investigation revealed that these difficulties are due to two main factors.
n Hastiness due to mindset
n Terminology issues
Sometimes hastiness can automatically occur due to the mindset that mathematics should be easy to follow even if you change the medium of instruction as you are dealing with symbols.
This attitude can cause enormous problems as students may skip instructions or avoid reading the question fully and concentrate only on the symbolic part of the problem.
As an example, consider the following question.
The graphs of lines 4x + 3y = 7 and 5x – 2y = 3 intersect at point A. Find the coordinates of A.
Seeing the word ‘graphs’ and the two equations, a student may be tempted to draw the graphs of the two lines and thereby find the point of intersection, which is a time-consuming affair. If it was read properly, the student could have noticed that the solution can be obtained by solving the two equations algebraically, which is much more efficient.
To a fast reader, obtaining the correct answer to the following question can be a problem as it may end up with just finding the value of x.
If 7 – 2x = 3x-8, find the value of 2x+5.
Therefore, it is important that the students are trained to read the question fully and understand what is required to be done, before attempting it. The students try to save time in different ways as they are also competing with time in examinations. However, it needs to be stressed that time spent to grasp the aim of the question is not wasted time.
Many children consider mathematics as an alien language consisting of symbols and expressions. Most of the difficulties that students encounter is related to vocabulary. The mathematical interpretation of the meaning of a word may differ from the meaning given to it in the English language. The word ‘find’ in mathematics means to obtain an answer showing the working while in the English language, it refers to discover or search. Consider the following example.
Find x given that x2 – 3x – 10 = 0.
If a student writes ‘x is the letter after -3 and before -10’, it may be considered correct according to the meaning of the word ‘find’ in English. However, in mathematics ‘find’ mean to figure out the value of x.
Two of the words that has caused much confusion are ‘or’ and ‘and’. In general usage, A or B is considered as either A or B but not both. However, in mathematics ‘A or B’ means ‘it can belong to A or B including both’, which is called the union. ‘A and B’ in general usage means the combination of A and B, while in mathematics ‘A and B’ refers only to what is common to both A and B, which is named as intersection.
Here are a few other words that can be misinterpreted with their mathematical meanings. The popularly accepted English definition is stated alongside while the mathematical interpretation is given in brackets.
Constant – Continuous ( A number that does not change)
Deduce – Draw as logical conclusion (To show a result using known information)
Determine – To get to a conclusion (Obtain the only possible answer)
Element – Component part (A member of a set)
Operation – Way things work (A procedure such as addition, subtraction, multiplication etc.)
Plot – Piece of ground or conspiracy (Mark the position of points on a diagram)
Similar – Having resemblance (Having the same shape but not necessarily the same size)
Volume – Power of sound or bound books (The extent of space occupied by a solid)
Write down – Record in writing (Obtain the answer – Working need not be shown)
The following illustrate some of the difficulties that the difference of meanings brings:
How odd these odd numbers are? The even numbers are even stranger.
Don’t be mean and help me to find the mean of these numbers.
Is right angle the right answer? Let me write it on the board.
The polysemous nature of some of the mathematical terms make it confusing for the students in the understanding of mathematical concepts. Mathematical terms have precise definitions to describe numerical relationships. At times these definitions resemble the everyday usage meaning but there are instances where the definitions notably differ.
Consider ‘in general’ as an example. In mathematics there can be no exceptions to a result if it is considered to hold in general. However, in everyday usage, if a claim is said to be true in general, it would mean that it is true most of the time, but exceptions are possible.
To add to the problem, there are some terms such as ‘degree’ that can have many different meanings within mathematics while having a different meaning in everyday use. In mathematics, degree can refer to the measurement of an angle, the complexity of an algebraic equation and a unit of temperature.
Although mathematics deals essentially with symbols, it is taught through the medium of language which is the major means of communication. Students build understanding as they process ideas through language. It is important for students to give emphasis to the familiarization with the mathematical vocabulary and at the same time understand the difference of meanings of terms mathematically and everyday usage. Teachers have an important role to play here in highlighting such terms and using them in different contexts for comfortable acclimatization. As Marcus Quintilianus quoted, “One should not aim at being possible to understand, but at being impossible to misunderstand.”
R.N.A. de Silva
ndesilva@osc.lk
The author is a senior mathematics examiner of the International Baccalaureate Organization and a member of the faculty of the Overseas School of Colombo.