Butchers, mobile phones and more. In this episode Andy and Adam discuss the great debate: Man or Machine. Why can't we always use calculators? What do they take away from learning? Plus, advice on what age that calculators should be introduced at school.

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Andy was one of the first to bring maths mastery to the UK as the founder and CEO of the independent publisher: Maths — No Problem! Since then, he’s continued to create innovative education products as Chairman of Fig Leaf Group. He’s won more than a few awards, helped schools all over the world raise attainment levels, and continues to build an inclusive, supportive education community.

With nearly 20 years of education experience, Emily has a knack for creating wildly successful learning content. Her past work includes publishers like Oxford University Press, Pearson and Collins Education. Currently, you’ll find her dreaming and scheming in her role as Head of Publishing at Fig Leaf Group.

In a past life, Adam was a headteacher, and the first Primary Maths Specialist Leader in Education in the UK. He led the NW1 Maths Hub’s delivery of NCETM’s Professional Development Lead Support Programme before taking on his current role of Maths Subject Specialist at Maths — No Problem!

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Hi, I'm Andy Psarianos.

Hello, I'm Emily Guille-Marrett.

Hi, I'm Adam Gifford.

This is the School of School Podcast. Welcome to the School of School Podcast.

Welcome back to another episode. Today, I've got something I think I need help with, right. So today, we're going to be talking about calculator: man or machine? And thinking about, well, what does that mean? And where does calculations sit anyway these days? But to make it easier for me, I'm going to pass it straight over to you, Andy, and just ask you to think about calculators: man or machine? Just open us up a little bit on that.

If you were to break this question down in a simple term, it would really be, should children have access to calculators when they're learning mathematics, in particular young children, and what's an appropriate age? And that's really, I guess, the main point. Like you highlighted, where does calculation fit in, in the real world nowadays? Is that a skill that we actually need?

So, when I went to school, and probably everybody before that, calculators were not an option, right? I think a calculator when I was 12 years old would probably have cost about a hundred dollars, which was an extraordinary amount of money in those days. Yes, I am that old. That wasn't an option, and most calculation was done by hand. Like if you went to the butcher, he would weigh stuff and he would mentally calculate in his mind how much owed him, or maybe scribble it on the package and then give it to you. And everybody did calculation that way, and it was an incredibly valuable skill. Who does that now? Right. Like, old people.

Cause they're the only ones who know how to do it! Right?

Yeah, yeah, yeah. And potentially things where you can actually do it faster than pulling your phone out of the pocket and sort of tapping in some numbers.

Yeah. If you buy three items that cost 20P, if you can't figure that one out, then you've got a problem, right? You shouldn't have to get your calculator out for that. But look, at some point in the curriculum, numbers get cumbersome and they get kind of in the way of the math, don't they? And I think that's really the issue. So if you're trying to teach a complex concept, there's no point in throwing in a whole bunch of tedious numbers that you have to wrestle with to get the correct answer, because you're taking away from what the construct is, with the object of the learning. If you're trying to learn geometry for example, it doesn't make sense to have angles like 37.8395 when all you're trying to do is discover what the internal angles of a particular shape add up to. If you're using angles like that, it's not helpful, you're taking away from the learning. I guess my question to you, Adam, throwing it back at you is, at what age should calculators be introduced in the school, do you think?

Can I dodge that ever so slightly? Because I just want to talk to something. No, no, no, I'll get back to it, but I think this is the right topic to address this.

I'm throwing you in the deep end there.

No, no, no, I'm happy to answer that. I'm absolutely happy to answer that. Because the reality is is that as soon as children have got a phone in their hand, which in my anecdotal observation gets progressively younger, that households with phones are now households with calculators. I wouldn't say that there's too many households that have got dedicated calculators, unless you've got children going through to secondary school or you've got them hanging around from previous schooling and whatnot.

But I think just one thing that's probably worth pointing out is how people see maths as well. Right? I believe it was a study, I think you're probably aware of this study. I can't remember if it was a Harvard study where it talked about the types of jobs that had the potential to be taken over by machines. A whole lot of information was put in in saying what types of jobs can be taken over by machines. Now you were talking about butchers and all sorts of people who are doing very quick calculations, many of them. But if we look at a job like an actuary who decides risk and puts a numerical value to risk, you know back pre-accessible computer days, there were huge calculations being undertaken in order to manage risks that did involve a phenomenal amount of calculations.

And these people, that's what they did, they calculated things and we needed it. I think, unfortunately, that might be an image of mathematics, and mathematics is far more artistic. And what the study showed is that jobs like actuaries and many other jobs that involved calculation, they were doomed. They will always be doomed, because in our pockets now, most of our phones could probably manage the types of calculations that used to take days. But interestingly in the study, some of the jobs with the highest likelihood of not being taken over, was a mathematician, because mathematician solve problems. So I think we need to address that. Calculation, we need that as a function, and there are some areas of mathematics that we need to undertake calculations in order to make something that we are looking at make more sense, or so that we can be accurate within those contexts.

And I think that that skillset can be introduced at any point, but I think that we need to be able to do things like add. So, I know one and one makes two. So when I introduce things like what if I had one quarter and another quarter, I'm not thinking about the calculation aspect, I no longer need to attend to it. I can go straight into understanding that when we add fractions effectively, we are adding nouns and we can just carry that out. But the calculation part doesn't stand in the way of the understanding of adding fractions.

When do we introduce calculations? Yeah. When it becomes too difficult to add mentally and it's just a bit of a waste of time because it's getting in the way of learning a new concept. And the numbers, they provide us practice, but I just think we need to be careful particularly into upper key stage two, or actually whenever a child has a difficulty with a calculation that should be something that they don't need to attend to. If that's stopping them from learning a new concept, we can address the calculation aspect of it, but I think we also don't want to put children at a disadvantage where a new concept's not being understood for lack of fluency in calculation. So, I've not really answered your question, but I think calculators can be used early, but not at the expense of being fluent to access those new concepts that come in quickly without the need.

So in practical terms, there are certain hurdles the children need to jump over and be able to accomplish regularly before they're given calculators, I think is kind of how I would interpret what you said. So, what do I mean by that? Well, if you want to be able to do any kind of calculation, there are certain things that you need to be able to do. You need to know your number facts up to 20 fluently, right? Like everyone always talks about multiplication tables and they don't talk about addition and subtraction so much. But actually, if you can't very quickly tell me what seven plus eight is, that's always going to be a difficulty that's going to get in the way of all kinds of things in your life. But beyond knowing all your number facts to 20, there's not much value in knowing what 37 plus 52 is.

When I say knowing, just remembering it is not really useful, because you can figure it out if you know all your number facts to 20. And that's kind of why 20 is that magical number, because of the base 10 system. So if you know all your number facts to 20, you can do all the addition to subtraction, as long as you also understand place value and a couple of algorithms, like column addition, column subtraction. If you can do those things repeatedly, successfully, then doing more of it isn't going to help you. Adding three digit numbers and adding seven digit numbers together, there's no difference, just that seven digit numbers is a lot more tedious. If you can add numbers with decimals in it, and you can add numbers that are three digits. That's all you need to know.

If you know that, if you know it confidently know it and can do it repeatedly, correct, there's no more value in doing any more of that. It's actually a waste of time. So at that stage, a calculator makes a lot of sense, because when you get into the more complex mathematics, who's got time? You don't want the children spending 20 minutes working out a complex calculation and not learning the topic that you're trying to teach, which is for example, like I said earlier, geometry. There's no point, it's not helpful, in no way, is it helpful.

And I think the other key point that you made, which is really important too is, what is mathematics? Mathematics is not arithmetic, like calculation. That's one element that you need to be able to do in order to succeed in mathematics. But mathematics is about something entirely different, and it's about generalizing patterns and problem solving at the heart. Being able to apply the correct heuristics to problems. When you look at a specific type of problem, you need to know how to work out the solution, that's the real thing about mathematics, you need to be able to work out, use the right heuristic. So different heuristics will be things like making a list. So if you've got a problem that involves things that happen in different intervals, or you've got two or more events, things like that, then a good heuristic to use is making a list, right? That's mathematics, knowing that that's when you use that heuristic, not knowing off the top of your head, what 328 times 4 is. What else do people need to know about calculators?

I think what's most important in the whole discussion around calculators, I'm talking about simple calculators here, right? Just simple, basic ones. We're not talking about scientific functionality, those sorts of things, but just the simple calculator, primary school. I think just be aware of what part calculation plays in developing a concept, and don't let it get in the way if you can help it. And just like Andy said, of course, if basic calculations are getting in the way of acquiring a new concept when children are older and we can expect children to understand this, then, yeah, of course, remedial work needs to be done. But if calculations are slowing children down too much and they can get there, they understand it, they know how to do the calculation, they're just maybe a little bit slower doing it, but it's in the way of acquiring a new concept, then give the children a support.

If we're not learning about calculation per se, then give them something that will help them out. If they understand the perimeter of a square is one side added together four times, if the calculation itself is tricky, then if the new concept's a perimeter, then yeah, of course, we can help out with the calculation itself. So yeah, I think that's the golden rule. I think that's the golden rule, just being mindful of those early calculations. Get those really solid, and then use them to your advantage when you get to new calculations, but don't labor it. That's the big one.

So there you have it. Give your kids some calculators.

Thank you for joining us on the School of School Podcast.

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