Hello everyone and welcome back to another episode of Teach
Sleep Repeat. My name is Dylan and my name's
Hayden, and this week we're going to be diving into our
favourite subject. You know what it is?
It's math. We're going to be talking
fluency, reasoning and problem solving, defining them and also
going over some of the misconceptions around them.
We'll take a look at how to structure a really quality maths
lesson and sequence of learning, as well as being able to use
schemes without them becoming limiting and frustrating.
And make sure you stick around, because we'll be giving some
sneak peek information about what we've been up to behind the
scenes in the Mass world. Oh, how exciting.
Let's go. All right, so I think before we
dive into this, we should just give some context.
Listeners will know how much we love maths.
They will know how involved we'll have maths and a whole
career. However, I just want to make
this really clear for anyone maybe who's watching or
listening for the first time or isn't quite sure about why we
feel like we can almost be the experts here talking about
maths, our careers. I for me, I think it's defined
by maths. Like I loved being a primary
school teacher because you could teach all the subjects, but my
passion and what I was really interested in and what I thought
I did really well was teach maths.
Same for you. I was a primary maths teacher,
Yeah. And I had to teach some other
subjects as well. That's, that's the way I see it.
My family literally asked like they got when I was just
throughout my career. They're like, Oh yeah, how how's
the old maths teaching? I was like, do you do know?
I do actually teach other stuff as well, even though I only talk
about maths. I do also teach English and
Geography everything. Else, yeah.
And I think when we trained the university, you know, our main
focus was maths as well. Like we literally had maths
specialist input from Gina Donaldson.
And Gina, I don't know if you're listening to Gina Donaldson, but
you are genuinely the reason why I felt like I could become an
expert in maths and I felt like I could pass it on to the
children because she inspired me.
Like she was unbelievable as a tutor.
It was at Canterbury Christchurch University, 10 out
of 10. And having done her course, I
felt equipped almost straight away because again, we trained
in 2014, 2015, around the time of this new curriculum.
So we were in this really odd position where in university we
were being taught how to teach maths in this almost new mastery
way, but embedded in the new curriculum from the start.
So when we went into school, we almost had a head start in that
respect, I think big time. And it was like, no, this is
just how you teach maths. And this whole idea of, oh, new
maths different to what I did before, we had none of that
really, because it was just that's how we were learned and.
Having now taught for 10 years, I can't believe how important
that was. I also had Gina so big up
transformational in terms of changing my just eradicating my
misconceptions about maths and how I very much thought of like,
yeah, it's just you see it online all the time now.
People being like, don't teach the kids multiple methods, just
do this one way. It always works and then they
can get the questions right in the test.
There are so many reasons I could argue against that and
we'll go into it and we talk about fluency and all the other
bits, but Gina really just immediately from from day one
was like, Yep, here is a more holistic way to think about how
people learn maths really well and and it just shaped my shaped
my maths and this. Is why it's better and This is
why it's good. And then when we went into
school, very quickly you became maths lead of of the school
where I worked. I remember you becoming maths
lead and I was on the hub, the team, but you were the leader
for that. And then when we ended up in our
school that we worked in together, just at the end of our
teaching journey, we both LED maths together, kind of Co LED
it. You're my assistant to the maths
lead. I was the maths lead.
So you were very much just kind of below me.
But yeah, we've LED maths in school for years and years and
years and years. And one of my big passions, and
This is why we're doing this episode, is to just help other
teachers understand maybe some of the terminology, fluency,
reasoning, problem solving, what it means, what it might look
like in a lesson, why it's important, which ones we should
weight more heavily, what it might look like in a classroom.
And importantly, what the misconceptions are as to what
good maths teaching actually might look like.
Which I think embedded in a lot of schools now for reasons where
they want to help their children as much as possible and want to
meet the national curriculum aims, they're actually doing
things that I think are potentially bad practise when it
comes to maths in general. So hopefully we can flesh some
bits out, talk about what different words mean literally
in this in the curriculum, and then yeah, dive into what a good
lesson might look like, how it might structure things and
basically let our passion shine, no doubt.
No doubt that's what you're saying, because we love maths
and we could just talk about this for hours.
Precisely that. So let's start off, I think it's
important that we everything we do is rooted in the national
curriculum aims. So I'm going to talk about the
aims specifically for the maths curriculum now in primary
school. And it's split up into three
chunks. And these three chunks are
things people will have heard of at some point in their training
or just in their delivery of maths lessons in primary school,
and that is fluency, reasoning and problem solving.
So let's read them through. Now this is from the national
curriculum. The National Curriculum for
Mathematics aims to ensure that all pupils, and that's really
important, all pupils, we'll come back to that number one,
become fluent in the fundamentals of mathematics,
including through varied and frequent practise with
increasingly complex problems over time so that pupils develop
a conceptual understanding and the ability to recall and apply
knowledge rapidly and accurately.
It goes on to then say #2 to reason mathematically by
following a line of inquiry, conjecturing relationships and
generalisations and developing an argument, justification or
proof using mathematical language.
And thirdly, can problem solve by applying their mathematics to
a variety of routine and non routine problems with increasing
sophistication, including breaking down problems into a
series of smaller, simpler steps and persevering in seeking
solutions. Excellent.
Shall we dive in then? Should we break these down?
Should we just go into fluency now and really just talk about
what fluency is? Again, some of the
misconceptions and just yeah, our top tips and what it looks
like in a lesson, how to embed it into lessons.
Yeah. Because when people say fluency,
I think you can sometimes simplify in your head as do you
know your timetables. Yeah.
Can you add up? Can you subtract?
And I think that's part of fluency.
I think that's really important. And one thing we'll be
referencing throughout this, I think it's a fantastic blog on
3rd Space Learning by Neil Ormond.
He did a whole blog about reasoning, problem solving, and
fluency. And a lot of what he says
resonates with the way I think we think about math as well.
So we'll be referencing that too.
We'll leave a link down in the description to that whole blog.
If you wanted to read, I would recommend it.
But fluency, to me, fluency it's being able to understand
mathematical facts and know them quickly, to understand concepts
and know them quickly and mathematical language and to
understand what they mean. And for me, being fluent in
something means practising doing the maths, understanding the
words or knowing the concepts. So it becomes second nature and
that's when you become fluent. So fluency is something you
should be working towards becoming.
Are you fluent in addition and subtraction at the year 4 level,
yes or no? The answer for most children
will be no. And we're working towards
becoming fluent. So when we say what does fluency
look like or varied fluency or mixed fluency in a maths lesson,
it's ensuring that children are exposed to ability to practise
things and not be scared of practising things over and over
because you need to be able to do the maths.
Do they understand the language being used in that topic?
And I would say being exposed to doing the maths, but in many
different representations. So I think a lot of people would
say, and maybe you can clarify a bit more about all this, a lot
of people would say that fluency equals arithmetic.
So just going a bit further from your, you said fluency might be
just times tables, but I think you can expand that to a genuine
misconception, which is that it's just arithmetic.
So do you think when it comes down to things like key stage 1
to arithmetic papers, or just generally arithmetic papers in
your ear group, do you think that embodies entirely fluency
because ultimately it's all the methods you need to know to do
maths? So what I'd say to people is
arithmetic is part of fluency. It's one string to the bow.
It's it's really, really, really important.
You could even say it's potentially the most important
part of fluency in my opinion. And I think we could argue that.
And I think you might be right to say that.
I think it's the actually being able to literally manipulate
number using the four operations is the most fundamentally
important part of maths. And then what I worry about
sometimes and what I see in schools is it's kind of brushed,
brushed aside or rushed through or sort.
Yeah, that's important, but we need to reason and problem
solve. No, no, no, you literally cannot
reason and problem solve if you can't do the maths.
So let's talk about arithmetic in particular.
I think that is vital and it very much links to the kind of
thing we're working on now. We're building platform to to
help teachers and children become fluent and specifically
to begin with get better at arithmetic.
Another sneaky yeah. Another sneaky is or.
Basically, it's true because I kind of want to talk about both,
because we're obviously extremely passionate about this
and I will, I genuinely think just backing on what you're
saying, I think arithmetic, particularly as part of fluency,
fluency in general, I would say is absolutely the most
fundamentally important part of maths, genuinely.
And I do feel like because we can also say the end goal of
maths is problem solving, I do think we conflate the importance
levels of them. It's like, yeah, the end goal is
problem solving, but fluency is the most important bit because
you can't do it without it. It's literally the foundation of
the house you're building. You have to get it right.
There's way more to it, you know, but if you think about the
classic pyramid example, I think even first place learning has a
couple of different examples of that.
It kind of is, you know, crudely, it's the massive bit at
the bottom, fluency, and that involves majority of that is
arithmetic. Yeah, we've got to get it right.
And I think that that's really important and that's the link.
We're going to be drawing these links throughout this and you'll
see why will maybe come back to it once we've defined, you know,
reasoning and problem solving. Why actually this is the most
important part. Fluency is the most fundamental
part of if you're teaching primary mathematics properly,
you have to get this right. And if you are glazing over it
to go through more quickly to reasoning and problem solving
without becoming secure in your fluency, you're doing your
children at the service because who can problem solve if they
can't once they've figured out what to do, actually do it.
It's the most important thing. And just going back to what you
said about fluency and we said arithmetics part of fluency.
So I can imagine people might be out there thinking, well, what
do you mean part? What's the rest?
For example, think of your part whole models, think of schemes
like White Rose learning, White Rose maths that a lot of a lot
of schools use out there and what questions might look like
1-2, three and four. And I can imagine sometimes as a
teacher, I've known teachers who become frustrated that, oh,
they've only done 4 questions like this, then suddenly they've
got a part hole model, then suddenly they've got a bar
model. Then suddenly they've got an
inverse operation and suddenly the presentation looks different
and the missing answer boxes in a different place.
That's down to you as a teacher. I think good schemes give lots
of opportunities for fluency like this.
But you as a teacher need to understand what fluency is and
why it's important and almost hierarchically move through
fluency in a way that's still sensible.
Because I think sometimes with fluency people here, OK, I kind
of get what you're saying now, Dylan, fluency is this like
ability to have lots of different representations of
questions and being able to apply.
Maybe let's just talk about addition to lots of different
ways. But oh, OK, even though this is
a pothole model, I'm adding, even though this is maybe a bar
model, I'm adding, this might be just literally written as
arithmetic, but I'm adding that is fluency.
But it doesn't mean you have to just randomly do them off the
bat. Big part of fluency is can you
do the maths in the 1st place? Can you literally add 2 numbers
together? OK, Then we can expand it to, to
some ideas of maybe, OK, look, maybe not just arithmetic now,
but it might be shown in a different way.
Yeah. And then there's even another
step of fluency, which is do you know how many sides the square
has exactly? That's not reasoning on problem
solving. Do you, do you know how many
centimetres are in a metre? These are.
This is when we talk about the known facts, the things you just
have to know. You just have to know the
definition of some words. You need to know what perimeter
means. That's not reasoning on problem
solving. It's not arithmetic either.
Exactly. It's not arithmetic, but it is
fluency and it comes under that umbrella and This is why fluency
is vital. If you don't know what words
mean, mathematical vocabulary, if you can't do the maths and if
you don't understand the concept, you can't do maths.
It's like, I can't get across enough how I feel like fluency
is maths and the reason people push back and they say, well, no
problem solving is maths, correct?
It overlaps. We can't see it as separate
things that we just do separately.
It's it's one in the same. And the way the aims of the
curriculum are written down and the way sometimes schemes of
work present things and the way sometimes we're taught how to
teach a maths lesson. I feel like fluency just gets
lumped in as like, yeah, you do that for a bit and then we do
some reasoning, problem solving. What I'm trying to get here is
with the definition of fluency is it is fundamentally what a
maths lesson should be and then you work towards so if you're
reasoning and problem solving which we'll.
Come on to moving through the the flow chart in my head when
I'm thinking about the three elements of maths, you know,
fluency, reason, problem solving is I think to myself, are the
kids fluent in the maths that I'm thinking of right now?
Yes or no? If the answer is yes, then I'll
start thinking, oh cool. I wonder if I can expose them to
some opportunities to solve some problems now where they can use
their reasoning skills and they can apply this fluency.
And if the answer is no, my genuine thought is either, OK,
cool, can I do some more fluency with them until they become
fluent? Maybe it's maybe it is just
straight arithmetic. If they're really, really
struggling even with the method at this point, maybe it's just
they're struggling with different representations.
They're not they've not been exposed enough yet to different
ways of seeing. They've got they're really good
at the method. They're just not very good at
recognising, Oh, that's when I need to use that method
definitions, definition equally, stuff like that.
That's outside arithmetic. And if and, but even still, if
the answer's no, I might still think, I still, I would like to
expose my children to a bit of problem solving, but I might use
something that fluent in, in something else, which we'll come
to, I guess, when we talk more about problem solving.
But the, the fundamental part of what I'm saying here is I always
think to myself, are they fluent or are they not?
You know, that's my first thought.
It's not just does the lesson struct to tell me I need to be
doing some reasoning on this objective today and some problem
solving. That's not the way to think
about maths because I they need to be fluent.
That's the point. If they're not fluent, don't do
it. It's such a good point and let's
move on to let's move on to reasoning now.
So we talked about, we talked about fluency.
Reasoning is, is to me really bizarre and unique.
Hardest one to define. I think so.
And I think, I think I can kind of define it, but I feel like
the old fashioned way of thinking of these as 3 pillars
of maths. I don't think it's helpful.
There's I feel like there's a very old fashioned way we think
of yeah, you've got your fluency, then you've got your
reasoning questions, then you've got your problem solving
questions. Yeah, sure.
And what I'm trying to do is we go through this, I think we're
trying to just eradicate that thought process.
Don't think of them as separate pots.
Because to me, reasoning is the ultimate version of you reason
all of the time, all of the time.
So reasoning again, let's think about what reasoning actually is
to define it. I'm going to be taking from Neil
Ormond in this in this blog because I think the way he puts
things down on paper is a really, really clear way of
thinking about it. Reasoning and maths is the
process of applying logical thinking to a situation to
derive the correct problem solving strategy for a given
question. Put more simply, mathematical
reasoning is the bridge between fluency and problem solving.
It allows pupils to use the former accurately to carry carry
out the latter. And I think that's such a nice
way of thinking about it is you can't just think about reasoning
as have I done reasoning questions today.
Reasoning is the art of thinking.
It's the art of of actually doing maths.
And I'd argue there's an overlap of fluency.
Clearly there's an overlap here with problem solving because we
have to reason and think about what's what we're going to pick
to solve a problem or come on to problem solving.
But also, I think with fluency, I think picking efficient
methods, that's reasoning, thinking, oh, what if I had to
explain to someone why this method is more efficient than
that method? That is definitely reasoning.
And it's the kind of thing I'd be doing during my fluency part
of my lesson. I wouldn't be thinking, oh
Christ, no, I can't possibly reason now.
I mean, my fluency part of my lesson, we're just practising.
No, no, reasoning is everything. And I think quality reasoning
happens to all children all of the time, no matter what they're
doing. Because at any point as a
teacher, you can be asking questions.
Why doing that? What's the benefit of doing
that? Why have you exchanged there in
that question? Yeah.
Well, that's weird because on
the other table they did this method.
Do you think that's better or worse?
They're still doing. Why have you used inverse here
Exactly. You know, even just that you
said. I agree, by the way.
I think reasoning is very much. It's the glue.
Yeah. And it does.
It ranges all the way from fluency all the way into problem
solving. Yeah.
I can think of the most basic version of this where you could
just where you could just about say, yeah, actually there's a
bit of a reasoning going on here, which is right.
You've taught the kids the arithmetic of adding two
numbers. They know the method.
They know how to line it up and do the addition and when
something says this, add this, they can do it, right?
And then you present them. This is all with influence.
You still based on all these definitions when you then
present them an inverse question where it's blank.
Take away a number equals a number and you know, as a
teacher, oh, they've actually, even though it says the
subtraction symbol in this question, they've got to add
those two parts together to get back to that total.
They're still using the exact same method you taught.
There's nothing different about the method and the arithmetic
that they've got to use to solve this.
But technically they've got to think just a little bit more
than the last ones of oh, wait, am I supposed to be subtracting
here? Or oh, wait, it's laid out this
way, which means I need to use inverse, which means I'm still
using addition. Great, I can do that.
That to me is like the smallest increment where I can think of
where it's like you could say there's a bit of reasoning
happening there now, but that's very much fluency.
So that kind of just backs up my idea.
This idea that reasoning really is the glue.
It happens all the way through fluency.
It happens all the way up to just complete problem solving.
It's kind of everywhere. It's like you said, it's not
really like a, a separate thing isolated.
You've got your fluency, then you've got this thing explicitly
called reasoning, and then you've got this thing explicitly
called problem solving. It doesn't work like that.
I think that the reason why I think this has happened and
developed over time is because I do think there are some
questions which are you can't refute that it's it's a
reasoning question. They do exist, like they they
can exist in terms of explaining your method.
You've seen these questions a million times in all these
schemes where it says, you know, think, think like Tiny from
White Rose maths, right? Tiny's done this.
Where has he gone wrong? Yeah, that's that's fun, that
is. Reasoning.
Explain your reasoning. Explain your reasoning.
It is literally reasoning, yeah, but again, you've literally
can't do that reasoning unless you understand the process.
Yeah, you've got to be fluent in the thing he's talking about.
Exactly how can you possibly do that?
And then that's when good teaching in the moment of doing
fluency, it is embedding these reasoning questions as you go.
Because then what the children can do when it comes to that
reasoning question is they think back to when they explain to
their teacher or they explain to their partner.
And the power of talk and the power of reasoning whilst doing
fluency means they can now access that question.
They are all interlinked. It is not a separate thing.
You should not be just getting kids just to practise maths,
practise maths, practise maths, practise maths.
Let's do some reasoning. No, no, no, it's embedded in the
very moment of learning the process.
You can have reasoning going on all of the time and the best
teachers do that. And that's where I think it's
absolutely right that we have these different names in the
curriculum. But I just hoping to get through
to people that you, you don't have to isolate them.
And if you do, I think it's at the detriment of your maths
teaching. So think about the last one,
problem solving. I'm going to start off by saying
to me, I said about earlier how fluency is maths, right?
Yeah, I'm going to say some people might think I'm being
hypocritical here or contradicting myself.
Problem solving is math hypocrite, actually, because I,
I think to me, if you, if you think about why we teach
fluency, right, it's not for fun.
I personally used to love adding up numbers when I was younger
because I just liked the patterns.
Yeah. I enjoy it, feels good to get it
right. Off it felt good and it's right
or wrong and I quite liked that. OK, and you could say oh, that's
fluency. No, no, the point of being
fluent and the point of reasoning is to apply this and
have a bridge to problem solving.
What is maths if not a way to understand and solve problems in
the real world? That, that is literally it.
I've got to catch that bus so that I get to my meeting at this
time. The bus takes this long, it's
only every 15 minutes. What time do I have to leave my
home? That's problem solving.
Yeah, I've got this much budget for the month.
I spend this much on my outgoings that I can't get rid
of. There's a grey area of ones that
I can. What can I get rid of and then
be below and have enough money to pay my water bill.
This. This is maths.
This is what I think school should be gearing towards.
I think there is a place for logical thinking that maybe
we're not going to use this in our real life.
So why are we doing that? Because it's a process of
problem solving. The idea of your brain solving a
problem and being logical is a transferable skill and even.
Within this moment you might argue you'll never use this
again, but you definitely will use the process of understanding
how to solve this problem. That's what problem solving is
for me, and it has to be a mixture of them and problem
solving. Let's define it.
And I completely agree with what I read about this, which was
it's easier sometimes to say what problem solving isn't so,
so we've got this here. So problem solving is not
necessarily just about answering word problems in maths.
If a child already has had available method to solve a
problem, problem solving has not occurred.
Problem solving in maths is finding a way to apply knowledge
for children who are fluent by the way and skills you have
learnt to answer unfamiliar types of questions.
That's the main bit, isn't it? That's the biggest bit.
I want everyone to focus on unfamiliar questions.
It's not technically problem solving if it's not a unique
situation for you to be solving this problem.
The question if it's if we're thinking about questions with
children, the questions that are being shown should be unfamiliar
to them. If you've modelled on the board
a particular problem and then what you've done is you've
changed the numbers and you've given them the same problem five
more times on a worksheet and said have a go.
But like sometimes what word problems do, it'll be the same.
It'll be the same method, but just different, different
contexts in the word problems that and you're saying this is
the problem solving part of the lesson.
It's not, it's not problem solving at all because you've
solved the problem for them on the board, which is what do I
do? How do I get from A to B?
How do I get from not knowing what to do to knowing the method
that solves it? And then you're just giving them
more arithmetic practise at that point because they know, they
know they know what to do because you've solved the
problem for them. So yeah, problem solving is A is
A, I'd say is the most like people get it wrong the most.
I think, I think I've spoke over my career, I've spoken to
teachers about maths and problem solving is the one where I find
myself being that guy that corrects people the most about
because they're like, Oh yeah, I've done some.
Problem solving Friday. Yeah, I've done problem solving
Friday. We do it every week, we do.
What I'll do is I'll have maybe 5 different problems, I'll model
just one of each one on the board for them and then they
crack on. So that's not problem solving.
Yeah, problem solving needs to be unique.
It's fundamentally part of what what it is, and you can't
problem solve unless you're fluent in the maths that
underpins it. I'm just going to add to what
you said there because it's so interesting.
I completely agree with what you're saying about problem
solving. Has to be new, has to be unique.
But realistically, there are points, certainly in problem
solving lessons, where things can't be completely unique all
of the time, right? If you are literally doing a
small step on additional subtraction, the kids may well
it's gonna be addition. Right, they know what?
I mean, so I don't necessarily think that in their situations
problem solving can't happen. I think it's harder and I think
there's still a place for modelling the steps in the
process of problem solving. So we talk about there like what
if you're at the front of the board and you model how to
answer this question, the kids go and do it.
It's not problem solving. Yeah, I slightly push back at
that a little bit in terms of, I think the way you model it with
everyone involved asking questions, reasoning, getting
the kids to explain themselves, thinking why we might try this
method or we might try that method that is vital in teaching
problem solving. So I think there is a huge place
for that. And you can absolutely say even
though my children didn't get 5 unique questions they'd never
seen before, therefore they didn't problem solve, I'd be
saying no, no, no, no, no. Because I think really good
problem solving teaching comes from really good modelling of
what problem solving is, involving your class in that and
then letting them have a go. And it might be that you model
them halfway there and you say right, you'd, what do you think
the last thing might be then? That's definitely problem
solving. But they don't necessarily have
to do everything from the start, all of the time off the cuff for
it to be good problem solving. I completely agree and I hope
that that I don't think you think that's a contradiction at
all, by the way. I think that's just an extra
point and that the easiest analogy I can make is honestly
to writing. Think of modelling writing.
It's exactly the same when you model writing to children, when
you expect children to do writing, by the way, you don't
just say off you go. Them modelling writing is
essential to them becoming good at writing.
But when you model that first paragraph or those first few
sentences on the board and they're all copying those
sentences or doing extremely similar sentences in their book,
because you've heavily modelled it, you don't count that.
But it's independent write, independent writing.
But the rest of it, you've shown them some skills that they can
transfer now and then when they're applying it to unique
situations, which is the rest of their writing or in math
situation, it might just be other problems now.
Yeah, that are equally. Yeah.
OK. It's all within addition,
subtraction. They know they're going to
probably be doing some adding and subtracting, but it's.
Got to work out what. The problems are still unique,
Yeah. How?
How much they're adding, How many times they're adding,
Whether they're subtracting, Whether there's going to be?
And context can be important. Different words, different
language. Exactly right.
Less than sold. Yeah, they they all mean
subtract. But this is still different
problem solving, because you're still accessing what words might
mean, because you're. Exactly right.
So I absolutely fundamentally agree you should still be
modelling two children how to solve different types of
problems, but also giving them a chance to independently solve
new unique problems. Even if you know that it's using
their the same problem solving skills but there's still unique
problems for them. Yeah, I want to, I want to.
We're going to, we're going to talk in a second.
I think more about how what that can look like.
If you're worried that your kids aren't fluent yet, can we talk
about that? Maybe we'll do some scaffolding,
but I really want to push on because I think it's something
that our listeners who maybe don't quite understand problem
solving completely in this way can really grasp onto.
And it's your comparison with independent writing.
I think that is such a good comparison because for a couple
of reasons. First reason I think about, I
think about what's the hardest thing for children to do in in
school, in primary school, because fundamentally adding 2
digit numbers together is quite easy and we can show that in
five to 10 minutes and children can practise it and they might
not get it straight away, but most.
Arithmetic is fairly straightforward.
Exactly that where and everything.
What's the hardest thing kids do in school?
It's solving problems in maths and it's doing independent
writing and I think we are so much further along on
independent writing in terms of it being more understood that of
course I'm not going to get my kids to write 2 paragraphs of
writing if they can't even do a sentence.
Of course I'm not going to ask my children to apply what a
front front of the verb you're writing in their work if I don't
already know they can do it and I've not introduced the idea of
adding information to them. This in independent writing,
it's so common that you're like, of course I'm going to model.
How can I possibly expect the children to have some really
good independent writing if I as the expert has not modelled to
them what it looks like. This is what problem solving is.
You don't just in your English lessons just say, ohh, by the
way, kids, we're just gonna do of independent writing now where
you need to workout what to do here.
And I'm, I'm going to be judging you based on whether you put
some, some independent clauses in here or not, or I'm going to
be looking for the this type of determiner.
No, it's very structured. You as the teacher are leading
them there and you let them have a go.
And if they get it wrong, it doesn't matter.
We're going to do it again. If they get it wrong, it doesn't
matter. We're going.
To put in opportunities to become fluent in those
fundamental writing strategies first.
But you've done a lesson. You've done lessons on
determinism. Whatever you said front of the
verbals. Yeah, that because you'll think,
right, They're fluent in these now.
And now hopefully they can apply it to this problem solving.
I mean, independent writing situation.
Yeah, it's so true. Isn't how English I think does
understand the process a bit more of fluency to the end goal
which is independent writing more than maths?
In understands fluency to the end goal which is problem
solving. I think they're a bit more
confused in maths. It can be, I think they can be.
I think, I think it's easier to understand.
I still think schools don't always get it right of
independent writing, but I do think sometimes it can be A tag
on a bit like problem solving is.
So we do a few weeks of learning and then we do an independent
writing every few weeks. I think both of them have to be
interweaved a lot more. And let's move on to now
something you said there, which which I thought was fascinating
about when to problem solve, because I think we can speak now
about we've got these three pillars.
And, and it very much felt like at the start of my career when
this was first being introduced because fluency and problem
solving, this idea of mastery and everyone doing like really
pushing themselves. I want to start by saying I
think it's vital. Every child problem solves,
every child reasons and every child becomes fluent in maths.
I think that's vital. But we're thinking about of
these three pillars, I don't think it should look like a 33%
split absolutely in your lesson, and I think that's bad practise
and dive into that and then we'll talk about what good
problems on my. Level Well, I even say that in
that split analogy, whatever the split is that you think is
optimal for your class, even that doesn't work on the
individual level. Different children have
different splits, genuinely, because different children will
require some children require a lot more time at the fluency
stage to become fluent. And if we, if we accept the
idea, this flow diagram of when you're fluent in something, you,
you can then start applying it to problem solving.
Then by definition, obviously some children take, it takes
them longer to become fluent in something.
So by definition, they're getting problem solving
opportunities at different points on that objective.
But there's another layer to this, because it doesn't mean,
which is the point you're getting at.
It doesn't mean then that those kids that take a lot longer to
become fluent, you can't just say, Oh, we've run out of time
again and no problem. They never get any problem
solving because we never get that fluency.
Finished yeah and didn't hate on the podcast.
Said I can't just give them problem solving.
They're just not getting there ever with the fluency is not
true. How do we solve?
What you can do is problems You can.
You can give them problem solving opportunities with
objectives that are not necessarily the exact unit
you're doing right now at the exact level.
It could be something from before you find something they
are fluent on. I've seen examples where
children are just maybe significantly behind their age
expected standard in maths. It doesn't.
They can't do any problem solving.
It just means you might need to drop back a year group or even 2
year groups and find something they are fluent in and provide
them problems that allow them to utilise that fluency.
So that's everyone gets a chance to problem solve.
It just might not be at the same level or the same level of
fluency underpinning that problem solving.
So you've got to find the balance.
It sounds almost slightly contradictory.
I get it because it's, I'm in one breath.
I'm saying keep going with the fluency until they're until
they're definitely fluent because then you can problem
solve on it. But I'm also saying, but do also
building opportunities to problem solve.
So there's got to be sometimes where you think, OK, we've done
a lot of fluency, but I am going to now build in, I'm going to
dedicate some time to do a little bit of problem solving
because I want them to be, I can't not do problem solving at
all ever. So I need to build in some time,
time and what that ends up transcending to sometimes in a
lot of schools, we've seen this over our whole careers.
People then go to the final version of this and they go, OK,
cool. So we'll put a policy in place
where everyone does a bit of fluency in their lesson and then
they do a bit of reasoning and a bit of problem solving.
If we do that and every lesson, don't worry.
Teachers, I know that children are at different paces, so I'll
let you do different amounts of them.
But everyone has to have fluency, reasoning and problem
solving in their lesson. Teacher A you can do 50%
relevancy, 15 reason problem solving, TGB.
You've got class find a bit tricky.
You can do 60% fluency and then just 40% reason problem solving,
but everyone's got to do it. Yeah, it.
Doesn't work, No. I think this is hopefully if we
are really establishing these definitions at the start, it
just becomes common sense that this can't happen and it doesn't
have to happen. And I remember as we were
preparing for Ofsted as maths leads master came in, we had a
very successful maths Ofsted and we get very successful data in
maths and all the schools that we've ran.
So we know this works and we were, we were looking up things
and it said like, it can actually be seen as bad practise
for problem solving to be shoehorned into every single
lesson. Because if you're doing that,
you're not doing problem solving for the right reasons.
You're doing it for coverage because you're worried it needs
to be seen. Problem solving can happen twice
a week. Problem solving can be can be a
whole lesson. It can be half a lesson once a
week. It can be every day.
It can be okay. It's not hard and fast raw.
And if you're thinking about what problem solving has to look
like and what fluency looks like and what we might see is like a
generic split. I think for me, the vast
majority of your maths teaching should be around fluency.
You have to get that right. You have to, it's as simple as
that. You have to teach the
curriculum, you have to get through things and children need
to be able to do maths. They were thinking about
reasoning as being able to understand concepts, procedures
and language. That is by far if you just think
about it for a second, you can't do the rest unless you have
that. So the vast majority of your
time in your lessons should be on fluency.
The reasoning happens all the time.
I wouldn't even, I honestly wouldn't even be saying to
someone, oh, you need to have a reasoning chunk in your lesson.
It would just be so built into my planning, to my questioning,
to my, to my fluency and to my teaching that fluency is
happening all the time in whether I'm I'm sorry, that
reasoning is happening all the time, whether I'm doing fluency
or problem solving. Reasoning is thinking like it
doesn't even need to be a separate thing.
And then I could either think to myself, OK, my children aren't
quite fluent in this. I'm going to carry on.
You might think, oh, my children are getting fluent in this, but
there's something else I want to recap and do and and keep them
fluent in that. And you might even think to
yourself, I think it's time for some problem solving.
And that might be my kids are fluent in this year for stuff
I'm doing. Oh, rub my hands again.
But let's try a problem where I know the cognitive load won't be
on the processes and the understanding.
So you're going to do it at your full?
Level. So I'm going to do at your full
level. However, I might well have a
class or even a group of children where I think, do you
know what you guys are fluent in Year 4, you're going to do it at
this year 4 level. You guys aren't.
But there's no way I can deprive your problem solving because
that's what maths is. It's that your name in the
curriculum. Let's just dip back a little
bit, but the art of problem solving, the ability to look at
and try and use your reasoning to know what to do.
The techniques involved, the types of question, whether it's
missing number, word, problem, finding all possibilities, trial
and improvement. Whatever the strategies might
be, everyone can use those strategies.
Everyone in any class can do trial and improvement.
It might be that ones in the context of something much more
simple for them to understand. It might be another one is much
trickier to know. That's what you've got to do.
But part of the problem solving is figuring out you got to do
that and then doing it. But you can fundamentally have
any amount of children with any level doing problem solving a
mass, and it's vital. And it doesn't have to be
shoehorned in. It doesn't have to be the same
amount of time every day. It just needs to be a whole
picture. And I think that's where you can
fall into that trap of tick box, tick box, tick box.
Because I want to just see that everyone's doing the right
thing. I want to make sure it's all
right. Yeah, it's got to be holistic,
because actually tick boxes by definition can force you to do
it wrong. Exactly what what happens a lot
in that situation, just to be extremely clear, is when you are
doing the tick box situation, you are then in a position
possibly where you're showing kids problem solving
opportunities and they're simply just not fluent in the maths
yet. And and therefore is it's a
complete waste of time. Because what you end up doing is
you end up modelling it all on the board, taking away the
problem solving element and end up inadvertently doing more
fluency, which probably realistically is not the worst
thing in the world because they clearly need more of that.
But it would be maybe more effective to have just been
carrying on with the way you're doing fluency.
A really good idea very quickly on problem solving, because I
love this as like a function is to to take the numbers out of
the question. Because sometimes by taking
numbers out of the question, you're kind of getting rid of
that difference there of like, oh, are you working at you 2
level, your four level, your five level of addition, let's
say? Because no, what that then
becomes is then the children are simply solving the problem.
And the question is simply just what do I have to do here?
What what is the, what is the like?
What do these words mean? Are you fluent in the vocabulary
we've learnt? Oh, you are brilliant.
That means you've been exposed to a bunch of vocabulary.
What do you think it might mean for a delivery to be shipped
somewhere? That is not, that is not the
word ad. That is not the symbol ad.
That is you think I'm really thinking to tell, OK, I kind of,
I can use my phone, become fluent in this.
I understand these words mean, I think we need to add, we don't
even have the number, but but the act of that and that this
can be reasoning and problem solving.
This is where it overlaps. You can explain to stuff.
I think it means add because the shop would have more of it.
Yeah. Oh, interesting.
What does the question say at the end?
The question says how much does the shop at the end of the week?
Oh, interesting. So is there something else in
the question that might happen in the week?
This is like as close as it gets to actually extracting pure like
problem solving and reason because you're almost taking out
the bit where they actually do the maths.
At this point. You can take my numbers.
I actually learned that. Hilariously, if we're begging up
people like Gina Donaldson, I actually learned that from
originally right at the beginning of my career, Gareth
Metcalf, who we had on this podcast two years ago now.
It's crazy to think that one of our earlier episodes, go and
find it. He's an unbelievably talented
mathematics teacher. He creates incredible resources
that I see reasoning, I see. I see problem solving as well.
Really, really good. But I first picked that up from
him. I went to one of his training at
many years ago. I was like, Oh my God, yeah,
just another, It's just another string to your bow, isn't it?
There's so many little things and tools and tricks and ways
you can expose children to isolated skills, as well as just
opportunities to just straight up problem solve, as well as
opportunities to just practise a method and arithmetic, as well
as opportunities for them to become fluent by showing them
five different ways of looking at an arithmetic question.
That's just ever so slightly different.
There's a bit of very fluency in there where they're using slight
elements of reasoning, but it's not problem solving because
ultimately it's just two numbers and you're either adding or
subtracting them or multiplying and dividing them.
You've got all these little levels and I just think you have
to genuinely to be a really good maths teacher, I think you just
have to be aware of all of these, these layers and elements
and aware of differences between reasoning, problem solving and
fluency and how they are, like you said, they are both pillars
in terms of the way we account for them and we, we, they're
three separate aims, but they're also not pillars in the idea
that they're completely separate things all the time, that they
flow for each other. Reasoning flows through fluency
and problem solving. And just having this grand
awareness of all these things in maths is what makes you a really
good maths teacher. And I think without these
things, what can sometimes happen is if you, it's the
classic, what I think a lot of people say that when I was at
school, this is what we did. We got wrote, taught a method
that works. When I see this number times
this number written like that, this is the method that works.
Just do that. Why are we bothering teaching
kids different methods? Is that what we're teaching kids
different methods? Because when the problem isn't
shown exactly like that, they then don't know what to do
because they were never really taught to think about it.
They were just taught. If you see this, do you can
teach a computer to do that. If you see this, apply this
method. It doesn't always work.
And to to just jump on that and finish off that's still
important to do exactly. It's still, well, not even
people who are saying why don't you just do that?
I was like, mate, I do. Yeah, that's what.
Fluency and arithmetic is yeah, that's really, really important.
You're not wrong. You are not wrong.
That is really important. But you know what I'm also going
to do? Just expose them to a different.
Way show them a different way. Exactly.
The most simple one I always think of right.
And this is something we're building into our, our best
software for maths that we're passionately building right now,
which is really, really focused at the moment on arithmetic.
And it will expand wider into fluency and more.
Yeah. One of the easiest things that
we've we've done is can you move the equal sign to the other
side? Because I can tell you in my
career how many times where especially earlier on where I
maybe didn't consider this, I'd show every single question as A
* B equals or A+B equals. And it'll always be arithmetic
and then equals at the end. And the kids genuinely thought
equals meant this is the answer. They thought it meant the symbol
where you put your answer next to it.
And they didn't really understand what equals actually
meant. And as soon as you then took the
exact same question, A * B equals and you put blank equals
a * B, they were like, oh, have I got Oh my God, what do I do?
Have I got to use? Have I got to use inverse?
Because because they don't know, they haven't been exposed to the
variety of fluency enough. They know the method and I will
practise that method. I'll make sure they are fluent
in the method itself, but I'll also make sure they're fluent in
a wider sense in this is some different ways to even represent
this arithmetic, Yeah. Yeah, like.
Precisely. And that is just fundamental to
the way we're building this app. This software that we're
building is like, it's not just methods, even though it's
unbelievably important. It's also here's some different
ways you might see the same arithmetic question, but still
applying the same method. And it's important kids are
exposed to it, by the way. So yeah, use this.
And, and the great thing is with this and fluency in general, I
think the, the really important thing about knowing facts, for
example, and becoming fluent at something is you are doing it
quite frequently as well. You're doing quite frequently,
not necessarily for ages and ages in terms of obviously you
need to teach a lesson and have varied fluency in there.
But then when when like I've maybe I've taught additional
subtraction and this kind of links in to be to be fair with
schemes and stuff in general. Which is, you know, it's very
common now to block things. Maybe we can talk about that a
little bit where you might block your learning.
And what we mean by blocks is there are essentially two ways
traditionally that you might teach maths. 1 is a spiral
curriculum and 1 is a block curriculum.
In a spiral curriculum, let's take additional subtraction.
Let's say you've got six weeks of that you need to do in year
4. In a spiral curriculum.
You do it for two weeks before Christmas, you do it for two
weeks before Easter, and then you do it for two weeks before
summer. And the idea being you come back
and maybe you get deeper each time.
The block curriculum would be you do all six weeks before
Christmas in a row because you're really diving deep into
it and getting those opportunities where for the
first week and a half I didn't really get a chance to problem
solve too much. But my goodness me, because I'm
doing it for six weeks, we're going to dive in.
They're going to become really fluent and get those
opportunities. I think block curriculum really
lends itself to be able to apply what we've talked about today
better. I think it's a good thing.
Truly mastering maths? To truly master it, I do think
it has a massive drawback, which again goes against is one thing
we've said which is fluency needs to be done all the time.
You need to be revisiting it all the time.
And that's where a block curriculum loses its power if
you don't do something about it. Because then those children
doing six weeks of additional subtraction might get really
deep and by November they are unbelievable, but they're not
going to do additional subtraction until next November
when they come back to it for the next year group.
So maybe you could talk talk a little bit about this as like,
why is it still important, even though maybe we have this idea
of mastery to become fluent in something and thinking about our
platform, even about arithmetic testing every week and becoming
really good at doing your maths. Why?
How can we square that with what schemes do now, which is?
Block so there's something really important that we need to
understand about block curriculums, which is they are
mapping coverage of your main lesson objective.
And I think sometimes it's easy to forget that that might not be
the be all and end all of the entirety of your maths lesson.
A really good maths lesson, and this is to, in my opinion, is
the best practise, is that there is a significant different chunk
of the allocated time that you have to teach maps that is
dedicated to your lesson objective and that very much be
following your, your curriculum map.
OK, your your block curriculum, but there should be a, another
significant part of your lesson, maybe smaller than the main bit,
which is dedicated to not that main lesson objective, but other
things. So mostly fluency.
It could even be problems. We'll come to that later.
But a part of your lesson is dedicated to recapping parts of
fluency that you've looked at in the past so that you're not
leaving at a year, year. It's we should never be in a
situation where you're actually leaving an objective an entire
year before it's remotely talked about or touched on again.
That is bad practise. And I, and I hate the idea that
anyone looks at block curriculum and says, well, that encourages
this, therefore it's bad. It doesn't actually encourage
it. It's really just mapping out
your main lesson objectives. You should be building as a
school, you should be building in time.
I think what do we have about 20 to 25 minutes.
We had this dedicated section, which is is nothing to do with
the main lesson objective. It could be, if you want it to
be, it could be, it could be linked, it could be we're pre
teaching a skill that we think that's going to be useful.
And by pre teaching, I'm literally talking about, let's
say you're doing rounding. And you know, if you, I would
say the same example, it's the easiest one to think of because
it's so like tangible when you're teaching rounding, like
obviously you need kids to know multiples of 10 or if you're
rounding the nearest 10, they they intrinsically need to know
what multiples of 10 are when you're teaching rounding.
So you might be doing a pre teaching in starters leading up
to this rounding unit where you're reminding them of
multiples of 10. If you feel that's what they
need, that's what I'd call as pre teaching teaching, but and
you've also got reteaching in that time.
It could be that you know what we finished this unit on
addition, subtraction. We use that example a lot.
And a month later, I just, I just wasn't satisfied fully that
they were fluent, as fluent as I'd like them to be.
And you know what, I'm going to use some of this time outside of
my main listing objective in later blocks across the year to
reteach some smaller parts of, of, of those of those small
steps, those elements of those lessons I'm going to reteach.
And the big one, the thing we do the most is, is just genuinely
just recapping. It's been a bit of time.
I want to make sure I'm covering and I'm going over and making
sure because fluency has to be done all the time and they're
building on it. I'm going to go back to old
objectives and I'm going to and I'm going to recap this.
Why is it difference to reteach? Reteach is I guess is from
mindset, right? So reteaching is specifically, I
feel like they weren't fluent, they weren't there, they weren't
quite there. And I and I want to make sure
I'm getting them to this a certain standard recapping.
They could be absolutely there, but it's very important that
it's still, yeah, refreshing their mind.
In that recap, obviously you're saying that they're still
fluent. Do you think in this, in this
like chunk? Because I used it a lot to be
like, cool, well, I can reason in problem Solver now as well,
yeah. Would that come into recapping
for you or like going over stuff?
Again, it would do yeah. So, so for me that is down to
all honesty like the the make up of your class as well.
So I've had classes before where genuinely, because we are
struggling with the fluency so much and yeah, I was building a
problem solving opportunities in the class, but fluency was a
much bigger issue than maybe other times.
That time for me became almost exclusively more time to recap
and reteach and preteach fluency things.
And very, very rarely, whereas I've had other classes where
because they're so fluent, it doesn't take me long to recap
things to make sure they're still up there.
It's like bouncing off the limiter of like, they're right
up there really, really understanding, really super
fluent. So I'd be like, cool, I've got
some opportunities here to to now recap or reteach some
opportunities from to problem solve as well.
Because actually that is always that is the end goal.
And I can and that's like, I can make that infinitely hard.
You can't complete problem solving.
It just goes on forever. So yeah, lots of different
things you can do. But my main point here, maybe
you can dive into it more, is you've got your main lesson
objective and you've got your coverage.
And as part of your lesson, you've got exclusively a special
part of your lesson that is designed for recapping, re
teaching, pre teaching, whatever else you need to do.
Yeah, I think it's vital. I think it's the way to make
schemes work like this. I don't think they work
otherwise. And I think I'm going to
slightly change what you said. I don't think it necessarily has
to be part of your lesson. I quite, I quite like thinking
of them as literally separate things.
I can think of my maths lesson when I'm going through the
curriculum that has the majority of that is fluency and it has to
because I'm teaching them how to do the maths.
I've had people message us saying I'm really frustrated.
My head teacher says every single lesson needs to have this
much reasoning and this much problem solving.
It's annoying me. I'm like, I I get why that's
annoying because it would annoy me because when I'm teaching the
curriculum, I have to teach you how to do the thing.
And part of that is going to be reasoning, problem solving.
Reasoning comes for everything. Part of it will be problem
solving. But the majority of my lesson is
fluency because I'm teaching you how to do the maths.
This separate thing here, the reason I love it so much and I,
I thought it was the best thing we ever did for maths in our
school was introducing this as a hard, hard separate thing.
Protected time. It's protected time.
It is literally protected time where you get 20 to 25 minutes.
And I used it for times tables because I was in year 4 big
focus on times tables. I did a lot of times tables in
there. I did a lot of recapping stuff
if I finished my block and I was like, I, I had a set that maybe
needed more support in maths, let's say.
And I finished my block and I thought, do you know what?
OK, I might sometimes choose to just do it for another week or
two. And that's also fine.
You've got to be flexible. But I might think, you know, I
really just need to move on actually.
So I'm going to move on. Lots of them got it, but some of
them didn't. Then I'd use that time to go
over that, become more fluent. Reteach some bits.
Literally, I would use a lot of that time to do some problem
solving because I found myself with my class who needed more
fluency. I really started to resent the
fact that I felt like I wasn't doing enough problem solving
with them in those lessons. So I thought, you know what I'm
going to, I'm going to make sure I do this in this time.
You nail the fluency in the lesson and.
Nail it. Yeah, I'm reacting to the kids.
Like there might be some lessons, like you said, where
I'm doing it all the time in my lessons because the kids are
fluent. My kids weren't quite fluent.
So I thought, you know what, I can do my extra time.
I'm going through the curriculum with them and I need to do that
in this time. I absolutely went on to the year
2 curriculum, downloaded PowerPoints of reasoning,
problem solving questions that based on what they, what they're
called. And I went, you know what?
Yeah, my kids absolutely can count that.
They can do that. That's fine.
They're struggling the year for objectives, but they can count
in twos. Let's look at some problem
solving questions that use counting in twos.
So my kids are no longer thinking, oh God, I don't know,
because I don't know how to do the maths and I'm already
thinking about these numbers are too big.
I don't really understand. No, no, the, the maths itself is
fine. They can put all of their effort
into thinking, what do I have to do to solve this problem?
And I, I found that chunk of time invaluable.
And we had to make it really clear to everyone.
That's 20 to 25 minutes a day. And a chunk of that would always
be practising known facts because you have to know, you
know, facts and that's fluency. And the chunk would be
reasoning, problem solving or preteaching like you just said.
But what was really powerful I found because it's what I would
have wanted, which is why we introduced it in our school.
It was all power to the teacher. Yeah, it was not.
I was not coming around and saying have you done your 10
minutes reason problem solving in this little section?
I couldn't care less. I said to the professional, as
the teacher, you are their teacher and you teach them 50
minutes every single day of maths.
What are you not giving them enough of?
What do they need more of? What do you decide?
If you had another 20 minutes, what would you do with the kids?
Yeah, yeah, yeah. Oh, that's interesting.
Do that. And that is your.
And my kids made so much more progress because I could be so
targeted in that time. And just just flipping that a
bit, people thinking, well, my maths lesson's shorter.
You just said your maths lesson was 50 minutes.
Yeah, because I wasn't obsessed with getting through reasoning,
problem solving and and doing my fluency all in one lesson.
It can be shorter. It's a very small step.
The curriculum can be condensed. 50 minutes is genuinely A
suitable amount of time. It is.
I remember thinking so short, remember going for an hour to 50
minutes once and I was like, Oh my God, is it possible?
How can I possibly teach maths? And it's because I was locked in
that mindset of I have to do XYZI, have to get all these
elements in. It's not a good maths lesson.
It's just not true. Just take them on the journey.
The whole lesson can be fluency. And I know there's multiple
years of my career where I felt like I would look terrible if
someone looked at my lesson. I looked at the book afterwards
and I'd only been working on fluency in that lesson.
And I wish I could go back and be like, no, if, if that's what
they needed because they were not ready.
And you felt like it wasn't appropriate to do problem
solving in this lesson yet. And you didn't want to go back
to the previous year because you're like right in the middle
of this little journey and you just wanted to finish off making
sure they're really fluent. Then that's what you need to do.
And that is OK. You didn't need to do that
reasoning, problem solving that you planned because you got told
to push it into a starter in another week.
Maybe you don't even use it or maybe use it in tomorrow's
season, whatever. But that's, that was the right
choice. You adapted on the spot and they
needed to become fluent. And there's one thing I will
just say, and we'll clarify this right now because it sounds like
we're being, I feel like we're, don't people think we're being a
bit wishy wishy of like, Hey, you just do everything ever,
ever, all the time. One thing I absolutely did just
nail and hammer all the time in my status was at the end of the
day, arithmetic. I would say at at minimum,
minimum once per week, my extra 2025 minutes was dedicated to
recapping or reteaching very specifically at the fluency
level, some arithmetic. And that yeah, that might
include, I've presented it in, in its inverse format or I've
put the equal sign on the other side just to give them that bit
of variety. But ultimately it was, I need us
to re practise this method because I taught this to you a
month ago. We haven't divided decimals by
numbers in a while because realistically that's one of 19
objectives by the by year 6. Here's here's 5 questions
dividing decimals. Can we remember it?
You've forgotten it. Cool. 22nd recap.
Because you were fluent. You were here before you were
super fluent. You've just dropped down a tiny
bit. I'm just getting you back up
there. Do you remember that?
Oh yeah, I remember that now. Brilliant.
We've practised it for 10 minutes.
Move on. So arithmetic embedded in in my
starters all the time. I did it every single day in
year 4, like every single day. It was, it was not even a
question. Like my kids got to the point
where they're like, oh, here it is the five questions again.
Because I do 5 arithmetic questions every day.
Some days it might just be addition.
Some days it might just be addition with different digits
because I know for a fact it's a misconception.
They can't learn at their numbers properly.
Some days it might be a mixture where I've said, actually, you
know what? We've done so much work over the
last couple weeks. Here are 5 questions with five
different question types. It's still arithmetic, but now
you're showing me your fluency and your reasoning because you
need to work out what to do each time.
It's not just doing the process to process, not reading it,
understanding the symbols. It's still a part of reasoning.
It's still fluency, but it's that really important skill that
children need to be able to do math quickly and accurately.
They have to be fluent. And arithmetic is the perfect
example of that. You have to be accurate.
And I'd say it to my children and I'd, I'd be saying it in a
way where like it's a really nice supportive thing where I'd
say to them, I know for a fact when we finish year 4, every
single person in this room would, will be able to, I say,
will will be able to answer every single question on your
repeated paper. You might make a mistake, that's
fine. You might get there and rush
through something or miss an exchange, That's OK.
But I know through my coverage and my daily practise and my
fluency that I know everyone in the room can answer that
question. And that's all I want to get to
guys. And then we'd be doing a
repetite at the end of the year, it's out of 40 and my class
would be averaging 35 and the odd mistake.
But there wouldn't be anything that a child goes, yeah, I've
never seen that before because I do it every day.
I'm not remember at all. Exactly.
And that's where this, this idea of our maths platform has kind
of come from. It's like, OK, how can we make
it easy for teachers to do this? Because I know the power of
doing arithmetic every single day and getting fluent in being
able to actually do the maths. But a teacher might not quite
get the nuances of the different questions.
They might not quite understand maybe why there's, you know, 2
exchanging in this question or going past the bridging past the
zero makes it harder, etcetera. And even if there are teachers
who get that, it can be time consuming and frustrating to
make sure you get coverage. And it can be time consuming and
annoying to get every individual child's data and be like, well,
that child can't do this, this and this.
This child can't do that and that.
But it's a different one as well.
They can't do. Our platform is going to be
something where you can really easily harvest this information
from the children. They can practise it.
They can do it every day. They get what they need.
It's adaptive to them. And it's going to be so powerful
to get children better arithmetic.
And it's something they don't have to do for an hour a day.
They can do it for 10 minutes. Yeah, yeah, yeah.
They can do a test, they can do a bit of practise on what they
need. And the teachers can go in and
say, actually I want this, this and this question.
Generate me 10 done. Yeah, it's it's going to be.
So I'm so gassed about it because anyone listening to this
who kind of gets what we're saying here about arithmetic, a
lot of people already message about it.
Message us and just let us know if it's something that interests
you because it's going to be so powerful.
And the reason we're doing it is not because we're saying teach
to a test. You just got to be able to do
arithmetic. Hayden, I'm hoping for what
we've said today. You can really understand that
unless you've nailed that, the true goal is pointless.
Yeah, You can't build on top of broken foundations.
You need to have the foundations in place.
We are trying to target the thing that matters the most,
which is the foundation. And we know one of the side
effects of mass mastery is the misconception of having to do.
We just need to put in loads more reasoning and problem
solving. And the horrible side effect of
that is in some places, in some schools, in some classrooms,
arithmetic and fluency is actually getting a little bit
weaker because what they would have been doing way more of that
before, they're now not doing as much because they think they
have to. I've got to do reasonable
problem solving every lesson. And now they're not as fluent.
And it's like, oh, man, we need it.
We. Yeah, we're trying to tackle the
problem from every angle. Yeah.
I I had a conversation with
someone the other day who said they're in year 6.
They messaged us and they said our school is thinking of
starting to do arithmetic every week and I was like, sure what?
They were doing excuse. Me and it's like people go so
far down this road of schemes and I suppose just wanted to get
your quick thought on schemes because I am not anti scheme.
I think schemes are really good. However, when implemented really
robustly and strictly and stringently and, and badly, they
can be awful. And I think this is a symptom of
this where schools might buy into schemes or say, yeah, we've
got this. It's, it's rooted in evidence,
It's a good scheme and it has lots of chances, fluency and
reasoning and problem solving, all that kind of stuff.
But maybe it's missing just enough to just practise and
become really fluent in actually doing the maths and then a
school doesn't do it because they're so wedded to the scheme
so very quickly. If you could just summarise our
thoughts on schemes. So now we're on the same page
here. Is that a reason to bin schemes
or do you just think, look, come on.
Absolutely not scheme. I'm going to say what I said
earlier. Schemes are really good for
mapping coverage of lesson objectives that you're teaching
the main bit of your lesson. Schemes are very good at that.
What I feel like schemes lack a lot of, and which is another
reason we're doing this, is because we feel like it's not
there. What schemes aren't so good at
is taking something like just arithmetic and providing loads
of resources and loads of fluency tools and ways for you
to just practise that bit and saying like, oh hey, in your 20
minute segment that you do in the afternoon or in your 20
minute segment that you do at the beginning of your lesson.
However, your school allocates this bit of extra time that
we're talking about. Like Dylan said earlier, it
doesn't need to be within your maths lesson.
It could be another part of the day.
Here's the resources and map for that.
And this is how this is some great.
It doesn't exist, I don't think not not to a not to a level that
I think is good enough, which is why we are building this
arithmetic tour. That is going to be I, I, I'm
getting too, I get too excited about have to calm myself down.
But I genuinely think it's going to be transformational because I
think if you get those foundations right with a really,
really good adaptive tool that works at the exact nuances of,
you gave an example. When you've got a tool that
knows the difference between a four digit number, add a three
digit number with exchange with exchange or exchange and the
different and all of the combinations of that and
understands those nuances. And if I want, if I want more of
these questions, it's like, yeah, you can have more of these
questions because that's what you want to practise.
Like you said in your starter, you might practise one day.
A very specific misconception that your kids are making.
Give them five more of those A. Misconception, which literally
couldn't be taught if you just did random numbers exactly
because if you're doing random numbers, you might not get
exchanging, you might not get the same number of digits,
etcetera. You're so right.
And it has it. This is where it's going to be
transformative because you can create 100 questions
regenerating the same thing. And do you know why I think it's
going to be such a good tool? Because it it puts together 2
things that we're really passionate about. 1 is quality
maths teaching and whether you are adept to it or not.
I'm not judging you as an individual because some people
are much more adept, like we know it inside out.
Other people won't and that's fine.
Their passion might be English, but they've got to teach maths
and they're really good at teaching maths.
Whether or not you're, you're really, really like to clued on
this can do it for you. And what it does is then allows
you to teach, this isn't going to be the thing that like
suddenly just takes over your class and teaches children how
to do it more than you know. What this is going to do is give
you the tools to be able to teach it so well and save you
time and give you exact things that the children need and do
gap analysis for you and give them targeted intervention.
It's going to do that. And then you as the teacher can
go around and say, right, you're struggling with this 5 minutes
with the white board, let's go. And you can teach them it.
It's so powerful. Can I fix because now we're
getting super excited. Can I finish by telling that
there's a, there's a moment that I can think of where I was like,
this tool just doesn't exist. This doesn't exist and I need
it. Which was it was it was a SATS
question that I was looking at. It was I can't the exact one.
It was something along the lines of like 2 1/2, lots of 57.
It was a really horrible. It was like the last question in
the SATS paper once. And I was like, that's rancid in
terms of giving the kids that and I wanted to have more.
I just, I was like, I want to practise these questions because
that's actually really hard. I've got a high mass set on a on
a practise mixed numbers times like really horrible numbers
where you can't even divide by the denomination.
Like 50 doesn't divide by two without going to decimals.
It's horrible. And when I used a random number
generator to just like generate more, they just, they didn't
get, they didn't get the nuance. It gave me stuff like 2 1/2,
lots of 20. And I was like, no, this isn't
what I want. I want it to understand so why
this is hard. So we're building this all that
does that and that's it. That's it for me.
I was like, I need to that needs to exist and that needs to exist
and it needs to get every single level of nuance on every
question and be able to just give me 10 more like that, you
know, anyways, but. You're right, you're right.
And I think this is it's built in.
Hopefully again from this whole episode, you get how much we
love maths. We're obsessed with maths.
We get good results for our children.
We literally, it's what our passion is.
And the ability to have been building this makes me so happy
because I'm not building this. And then, oh, yeah, it's
important because I've built this thing.
No, I, I, I believe everything I've said in this whole episode.
Yeah. And that's why I think this is
such a thing that I've thought for years would be so powerful
and good to build. It's happening like it's going
to be so, so, so useful. Watch this space, guys.
Yeah, keep asking us, because every time you ask us questions
about it, we get really giddy and excited.
So excited we're going to wrap it up, but there's just one last
thing I wanted to mention. And I don't think we need to go
into detail about it because we've gone into so much detail
already. But I think part of the problem
if we're going back full circle is obviously preparing for the
test, right? Yeah.
So think about SAT at the end and we're thinking, oh, well,
yeah, you're just creating a thing that helps you prepare for
the SAT. Yeah, it does.
It does help you prepare for the SAT.
It also makes kids better at maths.
But the Venn diagram can overlap, right?
Just because you're preparing for the test doesn't necessarily
mean everything you're doing is bad for the kids.
Fundamentally, knowing how to do maths and practising and getting
quicker and more accurate makes you a better mathematician.
Does it also mean you're going to boost your arithmetic scores
in your Sats? Yeah, Yeah, it does.
And and that's good as well, because you're better at maths.
Because you're better at maths, not because you're just knowing
how to get around a test or fudge your way through.
No, you can do maths more quickly and more accurately.
And one thing I was going to put over to you about the sats, you,
you're in your 6 for a long time.
I've never liked the naming of the sats.
Sounds ridiculous. There's arithmetic paper and a
reasoning paper. And I'm like, oh, what?
Like what? Like, OK, there are reasoning
questions, I suppose. Like explain this in the thought
bubble. Of course that's a reasoning
question. Fine, but the reasoning paper
has fluency in it. Yep.
It's not reasoning. It has just straight up fluency
questions like order these numbers.
Like it just has like how you know, how many sides does a
square have kind of thing. Like it has those kind of
fluency questions. It has some reasoning and it's
filled with problem solving, yeah as well.
It's, it has always annoyed me because I feel like it
contributes to the problem of like people getting confused
between the problem. Because realistically what's
happening is, yeah, you've kind of summarised it, the two
papers. It, it feels like it's supposed
to say we have a fluency paper and then we have a reasoning
problem solving paper. I feel like that's how people
interpret that when they see arithmetic and reasoning.
They're like, OK, that's the split they make in their mind.
But really the split is, yeah, we have this really a subset of
of fluency, which is arithmetic as one paper.
And then we have all of the rest of fluency, all of problem
solving, and all of reasoning that comes in between that,
because reasoning is just everything.
Because I I think I think there's some reasoning in
arithmetic. Some some people might tell me
I'm wrong here. What is the example I gave
earlier? Exactly like the the idea of in
arithmetic, the, you know, yeah, OK, being fluent includes the
ability to know which method to go for and why etcetera.
But but that is reasoning. Like it inherently is the
thought process of do you know what?
I'm in this test. I've got a method here that'll
take me 10 seconds or a method that'll take me a minute knowing
to go for the 10 second one. That's reasoning.
You're thinking it's. True.
I've got a really good example of that which I'll say really
quickly. Long division always happens,
right? If you, if you give a child A4
digit number divided by 45 and another question, it's a four
digit number divided by 47. A child that hasn't been taught
fluency properly with very fluency might just have one
method they use that works. And do you know what they could
get it right? And that's fine.
But the reason comes into play when you go, oh, some children
will look at 45 and think, I can factorise that into two short
divisions. I can divide by 9, get an
answer, divide that by 5, making it on the whole a lot easier
because I don't have to think about the 45 times tables and
finding huge remainders. That's way quicker to me.
And another child when you look at 47, they'll go, oh, that's a
prime. I can't, I can't factorise this
into into two times tables that I know.
So I'm going to have to just do this, my long division method.
And in the first one, they save loads of time, and the second
one, they're using the proper method.
And you're telling me they haven't done reasoning?
Yeah, that's what I mean. Like that.
Then that's in the arithmetic paper.
Exactly. So reasoning kind of is
everywhere, yet we call one of the papers reasoning.
I get it because it's more reasoning.
Arithmetic is more stripped. But it's not reasoning to just
be able to do a number that's adding ten.
But how often does that come up in the reasoning?
What's 10 more? What they do is to make it look
different though, is they'll put three of them in one question.
It'll be like this 1 + 10, this 1 + 10, this 1 + 10, this one
plus what equals this. You have to get all three to get
one mark and it's. Like each one of those questions
could have in the paper. Yeah, if it was just one of
them. So yeah, that's all I would say.
It's slightly confusing. I don't think it's an issue.
Like I said, I think reasoning permeates everything.
I don't think you can say it's not reasoning.
It's annoying, but I just think it just helps to push the
narrative of a bit of misunder. I mean, on the upside as well,
it does also just kind of reinforce the idea that
arithmetic is unbelievably important because they felt the
need to put an entire paper to that part of fluency.
Yes. And then call them, you know,
the naming we can, we can disagree with.
But yeah, ultimately arithmetic is like, yeah, it's essential.
It's an essential underpinning part of maths.
And we should be looking at can the kids do the maths and then
also looking at can the kids apply in any context, you know?
Absolutely got I, that is probably my favourite chat I've
had in a long time. There's.
So much more we could. Say, I know like somebody don't
see it, maybe this this one in the future.
But guys, if you enjoyed that, please do leave us a review
wherever you get it. And most importantly, honestly,
for this one, just share it around a little bit, especially,
you know, if you're a newer teacher, if you're training,
just send it around to people because it is really important
information to know. And if you're more experienced
staff, what I would say to you is what we've all said here,
it's rooted in, it's rooted in fact, it's rooted in research is
what it's what it is. So if you kind of are noticing
in your school, maybe the balance is a bit often and you
as a professional kind of have this idea of like it's not been
done properly. Push back a bit.
Like just just ask about it. Just say like look OK you might
get shot down fine but you might not and you might be able to
change stuff. And also talk to us because if
there's anything else more you want to, you want us to talk
about about Mouse, you know, we love it.
So if there's ever, if there's elements you thought, I wonder
if they're going to talk about this in this episode.
And we didn't please write in because there's almost
definitely another episode we'll do in the future where we can
talk about these other elements. Of maths, but we're going to be
building our maths platform every.
Episode from next week will exclusively be about maths this.
Is a maths, but it's not that we're going to change it.
Welcome back to MONEY. That toxic school soon fine.
Only if they're doing maths incorrectly.
See you next week. See.
You guys.