How technology adds to the learning process

I have just been attending a two day “Refresher CPD” course for Primary (Elementary) Mathematics Consultants in London.
We were asked to bring our laptops along with us and a few of us had them open and internet ready as Nigel, our speaker, began his talk.
We are an interesting selection of people, some of us are mathematics graduates and some of us, like me, come from a completely non-mathematical background and had given the subject up when at school because it was thought boring or just too difficult to understand.
This second group of people were “turned on” to mathematics, much later in life because we became teachers and had to teach the subject.We discovered that there was an order in the subject, an intrinsic beauty that had somehow escaped us when we sat in our secondary schools and were told that the quadratic equation was x= -b+ or – the square root of b squared – 4ac all over 2a (but were not told why this was and how it could be used in the real world).
The connection of the subject to other areas of the curriculum was not clear to us. I sat in many a boring physics lesson being told about velocity and how it was changed by forces working upon it and that this could be represented by a formula and worked out as an exact value. But it meant nothing to me! I learnt to use a log book (these were pre the days of a scientific calculator) and worked out a value that did not correspond to anything in my head other than a numerical answer that might be right and often wasn’t!
One very small thing in my mathematical education was to do with subtraction. I had been told that, in order to take away a unit that is larger than the unit it is to be taken from I would need to make my “subtractee” larger by 10 and pay this back by making the tens column of the “subtractor” number 10 more! It worked, so I never really queried it.. but for many years I did not understand why it worked and then, as an experienced teacher, I read a textbook for teachers on Primary mathematics that explained that I was making the “subtractee” larger by 10 and that, in order to keep the distance between the two numbers,as for example on a number line, the same, I would need to compensate the “subtractor” by adding 10 to it. (This is called the “Equal Addition” Method of subtraction) (For example in order to take 9 from 36 you start by calling the “subtractee” unit 10 more and now you have 16-9 , which is 7, but you will need to add a ten to the “subtractor” which makes it 30-10=20 and so the answer is 20 + 7 = 27!).
Now you may think, why on Earth do all that in the first place? The answer is… I don’t know! There are much better techniques available which we use now (and you can probably think of a lot of them yourself) but I wasn’t given the option as a child, I was taught this method and learnt to do it, without thinking.
But as I have developed as a teacher and later as a Mathematics Subject Leader for over ten years and for the last two years, as a Consultant advising teachers on the subject, I have begun to understand why much of it works.
In the Refresher CPD we were asked to take on some higher order ideas in the subject.This was for two reasons, to get us to further develop our skills in the subject and also to see how the complex can be made much more straightforward by explanation and, in this case, the chance to use technology to back up the complexity of the ideas.
So, on Thursday 14th January, in a lovely hotel in Gloucester Road, Kensington, London, a large number of consultants waited to be told about mathematical modelling and its relationship with the real world.
Nigel started by looking at Newton’s Laws. He did so by starting with a mini history lesson. We learnt about Sir Isaac as a rather solitary child whose life was effected very much by his father’s early death and his mother marrying a rather elderly country vicar who did not want to spend time with a child.
He painted a visual picture for us of this highly intelligent and questioning child, stuck in a dark, cold, candlelit room in Cambridge thinking about the nature of the universe. It was an excellent way to begin, because it made Sir Isaac into a person for me and therefore meant that I was now more open and responsive to what was to come.
We went through the first two laws fairly quickly. Then we looked at a simple diagram and worked from a simple model that he had attached to the wall. It was a model of a string attached to a flexible plastic bar. It made a form of triangle with the wall itself as the third side. When he pulled down on the point where the string and the plastic bar met we could quite plainly see that the string was under tension and the effect of pushing down was making the plastic tighter (or compressed to use the term that we would become used to).
He asked us to have a visual idea of a person jumping up and down at the point where the string met the plastic bar…. because he then showed us a new diagram where the original triangle was extended to four joined triangles. We were asked to consider which lines were “tensed” and which were “compressed”. Looking at the diagram we all began to see how this could be used in the real world and I remember saying to the person next to me “bridges”.
Having the laptop open I had been able to look up a Wikipedia article on Sir Isaac Newton whilst Nigel was explaining his life to us, I then looked up “Newton’s Laws” as he was speaking and finally, as we got onto the idea of why a triangular matrix can be used to provide strength for support in a bridge I was able to look up bridges and bridge design.
This interactivity with Nigel’s very clear and brilliant teaching made the whole experience very powerful for me and helped me to learn a complex idea effectively.
Contrast this to the bad old days that I describe above where a teacher would draw a diagram on a blackboard and then put in a lot of (to me) meaningless letters and arrows and then expect me to understand it.
I remember seeing a very interesting picture of students in an American University attending a lecture. The lecturer was talking to what seemed like row upon row of open laptops (you could hardly see the faces of the students). I thought this was rather strange when I saw it, but I don’t now. Those of us who had access to a laptop with an open internet connection were able to extend our ideas and thinking and make use of the huge amount of information that is available to students these days.
It is another example of where the technology is extending the learning for us all and where schools, colleges and Universities are withholding so much from their students if they do not let it become a routine part of the business of being a student

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