I’ve been inspired to write a piece defending maths from the old catch-cry, “I’m never going to use this in real life! And anyway, I can just use the calculator on my phone!”
While I understand and appreciate the humour, it’s an indicator of a deep-seated problem in western society. Many people were taught maths at school using methods that followed strict rules and any deviation from the rule – but which was equally valid at providing an answer – was punished. This resulted in fear of maths, and a fractured understanding of the underlying principles, leading to mistakes in logic. Unfortunately, those people are now parents, grandparents or teachers of today’s children, and though our approach to teaching maths principles has improved greatly (due to much research into this very problem), and results in far fewer misunderstandings, the fear remains and is communicated unconsciously. This is often made worse when parents cannot understand the new ways maths is being taught, and find themselves unable to help their children with even elementary maths homework.
I grew up in a maths-rich household, as my mum was a high school maths teacher (who incidentally went on to do a PhD in high school maths education, reached the pinnacle of her field in post-doctoral research, then retired; and now makes spectacular mathematical quilts which win awards and get cited in universities as wonderful examples of fractals – see patforsterblog.wordpress.com) and my dad was an engineer. It never occurred to me that maths could be scary, but then, I found it fairly easy to learn and enjoyed the process of following complex logic through to a single answer. I left high school very enthusiastic about maths, and have become fascinated by the teaching of numeracy in early childhood, because it is so much more complex than I could ever have imagined. And it is so rewarding to teach these concepts to little people, and watch them get excited when they realise the power of being able to see the answer and understand its application to the problem without having to use counters or their fingers to be sure their method works.
At this point, you’re probably saying to yourself, “It’s easy for her to say! She’s one of those weird people who likes maths. I never use maths in my daily life, and I’ve never needed it!” But guess what? You’re wrong. You use maths every day. Sometimes even without using a calculator. Come with me on a brief introduction to the foundations of numeracy, and you’ll see what I mean.
I’m going to use a simple example, one we can all answer, to illustrate my point:
2 + 3 = 5
Here are the concepts you need to know for this sum to make sense:
- Reading numerals: 2 when said aloud says “two”
- Linking numerals to their quantities:
- 2 means ♥♥- two objects
- 3 means ♥♥♥
- 5 means ♥♥♥♥♥
- Reading the symbols:
- + said aloud is “add” or “plus”
- = is “equals”
- Linking the symbols to their functions:
- “+” means how many altogether”
- “=” means “is the same as” or “the simplest way of expressing the answer is”
- Understanding why you use addition: “I want to know how many I have altogether if I know the size of the groups”
- Using the most common method for answering an addition question: “I have two counters in one hand and three counters in the other. I put them on the table and count them all.”
- Understanding counting:
- Saying number names in order
- Knowing that each object gets one number only
- Knowing that it isn’t important which object you count first, as long as you have included all of them once and none of them twice or more
- Knowing that the last number you say is the quantity you have
- Knowing that unless you remove some or add to the group, that quantity stays the same no matter how you arrange them
You may now be feeling the same way I did when I first tackled this at university. I was amazed to realise the incredible number of concepts required to handle a simple number sentence in such a way as you are sure you get the only possible answer every time. I think that we often overlook them because they are learnt so early that none of us remember it happening in our own minds. We all have pretty sketchy memories of our early childhood.
Now, here’s the important part. Imagine you’re a person who knows absolutely none of the above list of concepts. Somebody asks you, “How many places should I set for dinner? We usually have two places, but tonight we have three guests as well.” How can you possibly answer that question? You don’t understand the words “two” and “three”, you don’t realise that you need to take the two groups and count how many in both groups put together, and even if you knew that, you wouldn’t be able to get the answer because you can’t count anyway. Would a calculator be any use in this situation?
The next step is to use larger numbers. We spend time in school teaching shortcuts for adding larger numbers, but we rely on your understanding of the process of addition – getting all the groups together, and counting every piece – to make sure you get the right answer. Then we use word problems which simulate possible real uses for addition, such as the table setting one above. This is to make sure that you realise when addition is the appropriate operation to solve your problem, and not subtraction or multiplication. And, finally, we allow you to use a calculator. But you must always remember that the calculator will only answer the question you put into it. If you don’t understand the symbols, the concepts behind the symbols, and the nature of the problem, the calculator can’t help you.
Having taken you on this whirlwind expedition into 2+3=5, I’m sure you can now appreciate the volume of knowledge needed to solve simple, everyday problems such as setting the table, cooking using a recipe, doing the grocery shopping (for both quantities of food to buy, and money handling), organising a party, working out what time to leave the house to get to an appointment on time, or many, many more.
As well as this inherently useful way to organise and take control of our problems, maths is full of patterns – and human beings are suckers for patterns. In fact, not only do we love them and find them beautiful in themselves, we like to create them, and even have a tendency to see patterns where none really exist. This is evident in our ability to understand even simple probability games, such as the likelihood of the outcome of a dice roll. Although we can predict that the likelihood of rolling a 3 is 1 in 6, and that this probability is the same for each face of the dice, we still prefer to ascribe motives to the dice, and get surprised, and suspect the dice of being biased towards our friend, when he rolls five 3s in a row, even though each roll individually has the same probability of a 3 coming up, and the dice neither knows nor cares what the last rolls might have been. We enjoy patterns so much that we want to believe the dice has an affinity for one person over another to explain their seemingly orchestrated result.
Before I finish, I’d like to comment briefly on the reason we teach advanced maths in schools. It’s probably true that most of you won’t use calculus or the Pythagorean theorem in your lives once you leave school. But that is actually not important. The skills you acquire at school are not only meant to be useful in your daily life as an adult, they also make sure you develop all the brain function you need to cope with society as it is today. Humans have created mega-societies only rivalled by insect colonies in population, and these mega-societies require a huge number of incredibly complex logistical procedures in order to function safely and to everyone’s benefit. Learning advanced maths primes your logical reasoning skills, and you use those every day when you drive your car, judge when to cross the road, or write a chapter for your book.
Finally, of course, the products of advanced mathematics are all around you. You personally may not need to know how to do complex statistical analysis, but if no-one could, we’d have no cars, no medicine, no fridges, no planes, no computers, no mobile phones, no houses, no skyscrapers, no bridges…
Maybe it’s not such a terrible idea to make sure you have the basic tools by which all these things can be created. Maths is beautiful and logical and useful, and its evolution of complexity is guided by our increasingly complex and creative brains. Anyone for some maths?
Visit me at BuildingBridgesECS.com.au to arrange tutoring if your child is having maths difficulties.
Here are a few articles if you’d like to read more or find out about the research into this area:
“Being Numerate” from the EYLF PLP E-Newsletter
And in case you still need convincing, have a look at this classic Abbott & Costello sketch, which shows a possible consequence of not understanding basic operations!