x = 5y +3a
y = (x-3a)/5
I started learning about mathematical equations at the start of secondary school but it wasn’t until years later that I properly understood them.
I learned that when you move an item to the other side that it becomes the opposite. This was the basis upon which I endeavoured to do maths.
Then, years later, I had a moment of inspiration when I realised:
Ah………both sides equal. That’s why they’re called equations. Now I understand why you reverse items when they move to the other side!
4 = 1 +3
4 - 3 = 1
Looking back I feel utterly stupid for not twigging this essence of equations earlier. However, I came to feel less stupid when I pointed this same fact out to a couple of other people who immediately had a similar moment of illumination. So it wasn’t just me. I look back and think how easy maths would have been if I had grasped this essence of equations right at the start. When you understand the bigger picture of how something works then it becomes so much easier to work out.
Later, we were introduced to calculus. We were told how to do calculus by following a set of rules and processes for solving questions. But I never understood exactly what calculus was for. If I had been told at the start “differential calculus is about working out rates of change” and “integral calculus is about working out space between things” then I might have been in with a chance. I look back and wonder whether my fellow classmates either did understand it or were simply better at remembering and following rules. Maybe some did have a true grasp of the concepts and worked it out from there but I think most were just better at following rules. My memory is shockingly bad but (as I hope TheMaverickMan.com attests to) I enjoy thinking about the ‘bigger picture’ and I have difficulty learning unless I can relate what I am doing to the end result. I met a fellow class duffer on a bus several years later who told me that he had attempted A-level maths again and got a grade A – “it’s much easier second time around because you understand what it’s all about”.
Studying accountancy later on was like wading through treacle. How T-accounts work and how they relate to the Profit & Loss and Balance Sheet is baffling at first. My accountancy studies began with debit and credits - the small stuff – and then worked up towards the final accounts. In hindsight it would have been so much easier to start with the big picture and work backwards. If I were teaching accountancy the very first lecture would be to start with a personal balance sheet and understand that first before moving on, perhaps by then comparing a personal balance sheet to a set of company accounts:-
Money owed from Bob
Money owed to Fred
The balance sheet is Assets – Liabilities = Capital
or as many jurisdictions have it:
Assets = Liabilities + Capital (when liabilities goes to the other side it reverses – init simple!)
I never understood what the “capital” part really meant until I’d figured it out for myself much later. Sometimes it is referred to as Source of Funds which is substantially clearer. It is best described as “where the money has come from to get you to the current position”. If I’d been told that at the start then I wouldn’t have struggled so much in the beginning. It may have been mentioned but it really needed hammering home before moving on to anything else. Likewise understanding equations needed to start with everyone thoroughly grasping what they are for before attempting to actually do any. Simply knowing the bigger picture makes the detail much easier because you can relate each part of the process to the end result.
It is similar in the arts. I think teaching Shakespeare to teenagers, unless they are specialising in literature, is a waste of time. To appreciate the greatness of Shakespeare requires experience in life and more understanding about people than most teenagers are familiar with. Teaching it because the authorities think it’s something people should be exposed to is unlikely to endear people to it if a true understanding of it is beyond comprehension at that age.
People learn in different ways. Some people have good memories and I believe the most academically talented people are just good at regurgitating rules and methods. Some people need to understand the wider concept and yet this seems to be the very last thing that is ever prioritised.
The problem with teaching is that you, as teacher, are endeavouring to communicate to people who know nothing concepts which you understand like the back of your hand. Breaching that divide is the essence of being a good teacher. I can’t say I’ve met very many of them in my time.