The Importance of Units

Mathematics is meaning. As a teacher, I always insist that my students write their units. I do not force them to do this out of convention. Rather, I wish to impress upon my students that mathematics is all about meaning. Numbers are arbitrary without the units that give them meaning and describe their interaction with one another.

Meaning

Consciously or unconsciously we always attribute a unit to a number, especially when we are talking about quantity. “How many fingers do you have?” someone may ask. “10,” would be the reply. This, of course, means 10 fingers. So, while the unit was not spoken, it was definitely implied.

Units define what types of quantities we are looking at. For example, ‘cm’ refers to length or distance, ‘cm²‘ to area, ‘cm³‘ to volume, ‘ml’ to liquid volumes, ‘$’ to money, and so on.

Teach your child to derive meaning from the units they read and write. For example, consider this simple question:

John cuts 30cm of tape into 3 equal pieces. How long is each piece?

How would you explain this to your child? Instead of saying “30 divide by 3”, which is a common explanation for this question type, ask question to help your child think. Ask, “How long is the tape?”; and “How many equal pieces was it cut into?”

If I were to ask my students how they would solve this, I would not accept the response “30 divide by 3”, or worse still, just “divide”. You can’t find how long each piece is by “30 divide by 3”. You can, however, do so by “dividing the 30cm of tape into 3 equal pieces”.

Relationships

Units also show how two numbers relate with one another, especially in rates.

For example, consider “km/h”. Many students have the misconception that it means “kilometres divided by hours”. This is an erroneous definition. Rather, it means that for every hour, a certain distance is covered. For instance, a car travelling at 80km/h covers 80km every hour, twice of that in twice the time and half of that in half the time. I find that most students struggle with speed because they are taught to memorise the formula in the diagram below:

Now, it’s not a bad diagram, but it’s only useful in straightforward questions where only one of the variables is unknown. However, we know that that is not the case for many problem sums where students have difficulty even finding the correct variables to use. The formula won’t help them do that, but understanding units and their relationships will.

Reasonableness and Checking

In my opinion, the importance of “reasonableness” is not stressed enough.

Remember the uproar over the 2015 PSLE Maths question about the weight of 8 $1 coins? It’s a very valid question. Most people don’t understand that Mathematics has real world application.

I saw this issue in some of my P3 students. They could easily multiply 4 X 2. But when I asked them a simple application question: “There are 2 boys. Each of them for 4 apples. How many apples do they have altogether?” I got blank stares.

These are the same types of students that write answers like “There are 5.3 boys in the class”, “80cm” when asked to find an area, or “Jane has 9000 apples in her pocket”. None of those answers make any real-world sense, and a child that understands the importance of reasonableness and what units mean can easily spot these mistakes when checking their answers.

Ultimately, what differentiates a student that is good at maths from one that is great, is understanding – the understanding that Mathematics is not a discrete study of numbers and arithmetic functions with arbitrary function, but a study of the interactions and the relationships that bind all we see in the real world around us; the understanding that “mathematics is meaning”.

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