It seems to be a fairly common theme at the minute, what is the best way to tech maths. Normally I am most interested in teaching maths at a pre-university level but this article got my interest in how maths is being taught to younger students:
Now there a number of issues I have with the way being taught here, so I'm just going to look at two of the issues.
Before I get into the issues though I'd just like to note that what is being considered here is really serious stuff. I am a practising engineer and maths is fundamentally part of what I do. A lot of the work I do can be at the cutting edge of an industry but that does not actually mean the maths is hard. Most of the mathematical work I do is just basic geometry, force resolution and the application of linear equations.
So why is basic maths so important in modern engineering, surely we have access to computational methods to do highly complex calculations for us? Yes, we do, and yes we do use them. Every engineer having a desktop computer with their own suite of analysis tools is one of the biggest changes in the history of engineering. We can put numbers to problems now rapidly that have not been solvable throughout human history. It is a big change, but fundamentally basic maths is still important.
One example of this is in the basic checks of the results from a computer model. I once had to spend months solving a problem caused by a simple colon appearing in the wrong place in some code. Now, in pages and pages of code a colon missing is quite hard to pick up when you are self checking your work, but the reality is despite all of the sophistication that problem could have been identified by a basic hand calculation. A substantial proportion of the problems in engineering can be solved to about 10 or 20% accuracy in a page or two of hand calculations. This approach is fundamental to the application of a complex computational model. Perversely you usually should know what the answer will be before starting to build a computer model.
The other issue is that computer models take time and real world engineering often moves at a very fast pace. Most engineering problems are highly complex with multiple different inputs, some easy to put numbers to, and some very hard. All to often they are beyond mathematical optimisation and are too complex to consider every possible variation. Whilst we can put a lot of leg work into justifying one solution over another the reality is sometimes the decision about what the best way to do something can come down to a single meeting. And in that meeting multiple ideas might be considered which need to be quickly whittled down so you can focus on the optimum solution rather than spending time on solutions that are fundamentally flawed. Sometimes all it takes is a simple calculation to identify whether this an idea has merit or not and all to often a 'back of a napkin' calculation done in a meeting has been the deciding influence on whether an idea is taken forward or not. The problem of course is you have to get the maths right, your basic skills need to be up to scratch to make these important (often multi £ million) decisions confidently. Whilst a calculator might help, the reality is, in this environment it is your skill at manipulating numbers that is most important and those with a good grasp of basic maths are the ones that make the calculations accurately and confidently. Of course we do check things, especially in this environment, but basic skills are fundamentally important.
Okay, so what then are my issues with the modern ways of teaching. For this post my issues are all going to boil down to cognitive effort. The more things you have in mind at once, the more cognitive effort it takes. Have a look at the modern classic Thinking Fast and Slow by Daniel Kahneman to find out a bit more about cognitive effort. Fundamentally what it boils down to is if you have to keep too many thoughts in your head at once your ability to process them rapidly goes down as the number of thoughts and the amount of swapping of information you have to do increases. If you have to think about how to do the calculations then you are using up your limited amount of brain power. If you have practiced the basic maths skills enough to have internalised it you won't really have to think about it, you just get on and do the calculation leaving more mental effort for thinking about the important decisions you are making. It also means that you have more available mental effort to use in 'reviewing' the output of the calculation, so you are more likely to spot an error in the result. Anyone who thinks their maths is always perfect in engineering is kidding themselves.
The same problem applies if you teach too many different approaches to solving mathematical problems. Yes, I understand the arguments about people understanding this sort of problem differently, yes learning by wrote sucks. But the reality is, if you are solving real problems, you are usually having a difficult enough time without having to decide which one of six different methods of just doing the plain maths you need to apply, especially if some of the methods have limited applicability. Learning one method that works in all cases is always a better solution than many different methods with their own limitations. Again, it is all about cognitive effort.
Now, at this point, I don't want to appear to be too much of a killjoy. The one aspect of these new methods that I am a fan of is that they are attempting to put maths into context whilst teaching it. you are engaging students enthusiasm for something other than pure maths to give them the enthusiasm and drive to put the effort in to learn it. There are so many anecdotes about people who couldn't learn maths until they became a mechanic or took up darts but then that context gave them both the drive and the context to learn the subject. And this is what we need to focus on, giving students the right environment to learn the basic skills, rather than bombarding them with ultimately irrelevant methods to solve esoteric problems.
P.S. After writing this I read a few articles looking at this sort of subject, partly in a similar way, but also maybe partly in a different way. Reviewing my thinking now, it feels like I could be coming across as someone who wants to take the 'fun' out of learning and go back to the old ways of learning with thirty pupils all in rows learning by wrote from the teacher. I thought it was worth clarifying that it is definitely not what I would like to see. Of everything that I wrote on this post, probably to me, the last paragraph is the most important, you have to make something real and relevant and solvable to students. Given them a scenario, and probably just as important don't pigeonhole the work that they do. Let them solve the problem in ways that they think of. What you do need to do is guide them in appropriate directions, but don't force them to solve a problem by maths if they find a way of solving by building model. Yes, ultimately there are some fundamental ways of solving a problem that they will have to learn, but give them the time and space to work something out for themselves. You also need to instill qualities like craftsmanship and precision in the work they do.
So in support of my thinking I off up the following articles that are also well worth a read. I feel there is a common thread to this and one common item that has to occur in ever case - good quality teaching. A teacher that can work with the pupils and the path they are taking and that can work with the pupils enthusiasm and imagination, to come up with scenarios that will engage them and ultimately make the learning they are doing very real.
Here are a few articles that are well worth a read and that I think support this view.