Maths: Conceptual understanding first, or procedural fluency?

Should you teach conceptual understanding first, or focus on raw procedural fluency?

This question drives endless debate in maths education, but its answer is very straightforward: it depends.

I can demonstrate this quickly and easily with a single example, by teaching you how to multiply logadeons (e.g. 5-:-9,) something you’re probably not familiar with already.

Observe the following examples:

8-:-20     *   2-:-5       = 10-:-25

9-:-20     *   2-:-5       = 11-:-25

100-:-50 *   30-:-7     = 130-:-57

19-:-20    *   5-:-5      = 24-:-25

By this point, you can probably multiply logadeons together quite comfortably.  If you’d like to give it a go, try these two (answers at the end.)

30-:-17  * 4-:-3

17-:-0.5 * 9-:-2

But even if you can evaluate those correctly, you’re probably still not comfortable about all this; you probably don’t feel like you understand it, and for two reasons:

The first, is *why* does multiplying two logadeons result in us adding the digits?  It’s reasonable to assume there is a perfectly valid reason, just as is the case for adding indices when we multiply numbers in index form, but we don’t yet understand why it’s the case.

The second, is what the hell is a logadeon anyway??  I’ve spent a hundred or so words now discussing the multiplication of something that probably feels like a mental black hole in your mind; it’s very difficult focus on the process, and leave behind that question: “What on Earth is a logadeon?  What on Earth is he talking about?!”


Let’s switch to multiplying fractions instead.

If our goal is to teach pupils how to multiply together two fractions, then I would argue they first need to understand what we mean by the word ‘fraction,’ otherwise we’re saying meaningless things from their perspective (they also need to understand something of what we mean by the word ‘multiply.’)  This is to say, they need to understand fractions as a concept.  Arguably, they should be able to conceptualise fractions as parts of a whole, as ratios, as representing division of two numbers, and as positions on the number line, all of this, before we try to teach pupils how to perform arithmetic operations on this concept we call ‘fraction.’

Assuming we’re successful in that, we now wish to teach them multiplication.  Traditionally, we might simply explain that we multiply numerators together, and denominators together, and nothing more.  There is a reason that process works, and it can be explained in several ways.  It is also possible to relate the arithmetic process of multiplying fractions to other conceptualisations, such as a visual representation.  Here, we run into problems, though.  I’ve heard people argue that this visual representation offers an explanation, a why for the arithmetic process; it doesn’t.  It offers no proof, therefore no ‘why,’ simply an alternative means of conceptualisation, or to think of it differently, it’s another process for multiplying fractions, one that is significantly less efficient than the arithmetic process.

Visualisation of fraction multiplication

Now, is the proof necessary?  Are the alternative conceptualisations necessary?  We *must* conclude that no, they are not.  We must conclude that no, they are not, first because pupils can comfortably learn to multiply fractions without them, but more, because there is an almost infinity of proofs and conceptualisations within mathematics that can be related to the things we teach, we cannot possibly teach them all, and we therefore cannot possibly deem them all necessary.

But are they desirable?  Absolutely, and this is an important clarification.

Where something is necessary, such as developing the understanding of a concept before we move on to discuss what we can do with that concept, we have no choice.

Where something is desirable, such as providing proof for a process, or alternative conceptualisations, we have a lot of important decisions to make, and we must make them with careful and deliberate intentionality.

Often, standard processes are simple.  Proofs are relatively complex, as are alternative conceptualisations.  Try proving why the formula for the volume of the pyramid works, for example, which requires calculus.  Things get even messier for the cone, and sphere.  Deriving the quadratic formula is far, far more complex than using it to solve quadratics.  Pedagogically, it’s very difficult to turn these proofs into cognitive work for pupils, as well, making it tricky to fit them meaningfully into lessons (though certainly not impossible.)

In conclusion, there are times when we have to teach the concept first, and there are times when we often do not pay this close enough attention, jumping into processes before the concept is locked in place e.g. vectors, equations, irrational numbers… almost everything really.

There are other times, however, when this is being conflated with proof and alternative conceptualisations.  These should usually probably come after fluency with processes and algorithms is achieved, since those processes are often more simple than their derivations.  They must also be selected and taught carefully, since we have finite time: which proofs and which alternative conceptualisations will most help pupils develop a relational understanding of mathematics?  And when in a sequence of learning – which can take place over years – is teaching them going to be most successful?



30-:-17  * 4-:-3 = 34-:-20

17-:-0.5 * 9-:-2 = 26-:-2.5

And logadeons are not a thing, but sometimes it’s good to remember what it’s like to learn something without pre-existing expert knowledge.




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Never ask pupils a question to which they have not already been told the answer.

Never ask pupils a question to which they have not already been told the answer, unless they know enough that answering the question requires them only inching forwards.

Years ago I wrote on questions and questioning, a seemingly important aspect of teaching.  For anyone interested, here:

28th May 2013

29th May 2013

16th June 2013

15th March 2014

Which is really all to say that despite the irreverence, three years on and the question of questions hasn’t disappeared.

At this point, I would say they are a vitally important part of teaching.  During my training I was told that they were a vitally important part of teaching.  So where did it all go wrong?  Our understanding of the role of questions is flawed.

Consider the following two views:

1 – “Never ask pupils a question to which they have not already been told the answer.”


2 – “Use questions to ‘move pupils’ thinking forward,’ or to give them a chance to ‘apply what they have learned.'”

The current status quo is focused on Point 2, but does it badly, leading to bad results.

Point 1 is immunisation from questions such as:

“What do you think we mean by Globalisation?”

A question I’ve seen posed to a Y12 BTEC class.


“What is a revolution? What revolutions have you heard of? What might be the key features of a revolution?”

Posed to a Y9 class *before* being taught anything about revolutions.

These are both highly typical, but terrible sets of questions.  I’ve discussed them with teachers, and can say that people really, really think that these are appropriate questions to use in educating.

They stem from the view that we need to ‘explore what pupils know,’ or that pupil voice matters. It’s true that prior knowledge is the greatest indicator of future success, and that pupil voice in a lesson can be important, but these kinds of questions:

(a) aren’t the best way of evaluating prior knowledge *and*

(b) given the context it’s probably simpler and more efficient to assume no knowledge, and try simply to anticipate prior knowledge that might interfere with current understanding

With respect to (b), a good example of this for me was in learning about the Carnot Engine in Year 1 Thermodynamics. The lecturer didn’t quite set up the introduction well enough, so when she started talking about this hypothetical heat ‘engine,’ I struggled to dissociate it from the kinds of physical mechanical engines we’re already familiar with in everyday life, like in a car.


A Carnot Engine is absolutely *nothing* like this

So there *is* some important need to estimate the kinds of prior knowledge that might interfere with future conceptualisation, but, again, as in point (a) these kinds of questions are not the best way to do it (I’m not going to go into what is.)

Point 1 above is therefore powerful inoculation against this kind of sloppy thinking – if you’re going to ask pupils a question, make sure you’ve first taught them the answer.

Problem: following this line of reasoning, all Q&A becomes ‘factual regurgitation;’ to use less loaded language, there is no opportunity for pupils to try to generalise or apply what they’ve learnt to novel contexts – this would be a limited form of education.

So Point 2 above *is* necessary; the real question is how do we find the line of demarcation between when Point 1 is valid, and Point 2 becomes valid.

For this, variation theory in mathematics and ideas such as those from the Michel Thomas and Pimsleur language courses become a source of inspiration. These all rely on:

  1. Telling pupils explicit facts
  2. Asking them to recall those facts in response to questions
  3. Then carefully moving them on to something that hasn’t been previously taught
  4. But which is eminently within reach of their minds, given the new knowledge.


Michel Thomas

Voy‘ means ‘I am going,’ and ‘a‘ means ‘to.’

How would you say ‘I am going to?’ (Voy a)


If 2x + 5x = 7x, what is 2y + 5y equal to? What about 5x + 2x? 5x – 2x?

How do we apply this to modify the kinds of bad questions I noted at the top?

History example:

Do lots of work explaining what revolutions are. This can come in many forms, including teacher talk, reading, lists of key features, knowledge organisation, fact systems (see Engelmann), comparative case studies of situations that are and are not considered to have been revolutions.

In terms of questions, we now have two forms:

(1) Having studied the French revolution, ask pupils to explain what made it a ‘revolution’ (‘regurgitation of facts,’ or more precisely, responding to a question with the answer they’ve been taught – recall / testing effect)

Then later

(2) Give them something about the Russian revolution to read, and then ask whether is was a revolution or not. Or, the industrial revolution.

(I’m not saying these are great examples considering the structure and constraints of a real school curriculum and time in class, and my limitations as a history teacher!  I’m just using them in an attempt to exemplify the theory)

Globalisation example:

In this case, preempt in speaking to pupils that they have probably heard the word before, along with some of the things they *might* think it means; explain that there is much more to it and that it has some technical specification beyond how we use it in everyday speech; explain these features as above with revolutions; then go into questions around the fuzzy boundaries ‘Are these a feature of / or caused by Globalisation?’ ‘How will Globalisation impact on that other thing we previously learnt about?’ etc.  The challenge is in ensuring enough knowledge has been previously embedded that these don’t become questions that require ‘guessing,’ but rather require only a small leap in logic.

Perhaps this is a good summarisation:

Never ask pupils a question to which they have not already been told the answer, unless they know enough that answering the question requires them only inching forwards.

This is worlds apart from ‘guess what’s in my head,’ sprawling ‘what do you think,’ or ‘who knows how to do this and can tell everyone else, so I don’t have to’ style questions.

It’s hard.  Really hard, to do this and get it right.  The goal in variation theory and the language courses mentioned is not to ‘explore’ what students think about the language, but to help them connect prior knowledge to new knowledge in a novel context (whilst leveraging the retrieval effect.)

The goal is generalisation, transfer, flexible knowledge.  In these programmes, students should always be able to respond correctly to the questions based upon what they have been taught before, and getting that right is damned difficult; it places all of the responsibility for pupil success viscerally on the shoulders of the teaching.



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Why we need to get rid of lesson objectives

The problem of Juxtaposition Prompting

In sum:

1) The problem of Juxtaposition Prompting is endemic in our classrooms. It prevents generalisation and transfer, and therefore what we consider ‘deep understanding’ or ‘deep thinking.’

2) To overcome it, we must reconsider old lesson and curriculum structures, to carefully introduce greater variation into lessons, which will require us to remove lesson objectives as we know them.

Give this a go, if you like.



When I tried it three thoughts came to mind. First, this can definitely be solved using trig. Second, I wonder if there’s a simpler way to solve it using ratio. Third, I wonder if there are other ways to solve it that haven’t occurred to me.

Now let’s think about where this can fit into a lesson.

If we include it in a lesson about trig, pupils will automatically think they have to use trigonometry to solve it, and try that.

If we put it into a lesson in ratio, they’ll think they need to use ratio to solve it, and try that.

If we put it into a lesson on cumulative frequency diagrams, they’ll be deeply confused and refuse to engage.

This is the problem of Juxtaposition Prompting 1. We narrow pupils’ thinking, constrain their ability to generalise and prevent transfer by teaching pupils that they can use some feature of the lesson to figure out how to respond to our questions: ‘This is a ‘Pythagoras lesson,’ so there must be some way of using Pythagoras’ Theorem to solve this problem.’

In principle, the solution is to very, very carefully introduce greater variation into each lesson. In one moment I’m asking you to calculate an unknown angle, in the next I’m asking you to calculate the missing numbers in a set, having given you the set’s mean and range, and in the next I’m asking you to add three fractions together.

This is the same as what Bjork calls the Desirable Difficulty of Interleaving 2 (though I personally tend to prefer using that word to mean something related, but different,) and it’s noted as the Variability Effect by Sweller 3.

Although messy and confusing language, the concept of Juxtaposition Prompting helps us to understand why interleaving in this way can be useful, and why the Variability Effect manifests.

I used to run daily negative addition / subtraction drills with children in Year 9. The sheets always looked like this (credit to Bruno Reddy for creating the auto-generator in Excel)


Now, they *were* very effective, perhaps because I added in a few other bits and pieces around the edges (e.g. talking about what to do for each section and why, over time) and spent so very long on them – but it would have been more effective if they weren’t always arranged so neatly into the same columns.  The problem of Juxtaposition Prompting rears its ugly head here because pupils can, and did, quickly learn that ‘in column 2 I just add the numbers’ rather than ‘when subtracting a negative I can add it as a positive.’ They were able to respond correctly to the questions by attending to something extraneous to what I wanted them to learn.  I was intuitively aware of this problem and wanted to change the sheets to avoid it, but it’s a nightmare to create that kind of flexibility when you only have Excel available to program with; likewise, while it didn’t have a name before now, I’m sure most teachers reading this will have realised this problem before.

So we need to design worksheets and even whole lessons to remove pupils’ ability to preempt how to respond on the basis of ‘the lesson objective;’ the greater the variation in question type for any one lesson, the more we tear into this problem.  But I noted above that this must be done very, very carefully.   At the other end of the scale lies cognitive overload: too much variation, of the wrong kind, and pupils have no idea what’s happening or how to respond to any questions.  The trick is in successfully manoeuvring pupils to attend to the right things – to kick System 2 thinking into gear, if you like – without leaving them frustrated and feeling overwhelmed / overloaded.

Ultimately, pulling this off requires completely rethinking how we plan for learning over time. It requires us to abandon ‘lesson planning’ as we know it, with its objectives and careful A leads to B leads to C structures, and instead have many micro-objectives being studied and revised in every lesson. Engelmann’s curricula work much like this – referred to here as a ‘Strand Curriculum‘ design 5.

Strand 1

The black is a traditional Spiral Curriculum Design.  The grey shows a Strand Design

Strand 2



Scheduling the lessons in which a given objective will appear, over time


In these curricula objectives are not covered in a lesson, but over 50 or 100 lessons or more, simultaneously alongside other objectives.

In sum:

1) The problem of Juxtaposition Prompting is endemic in our classrooms. It prevents generalisation and transfer, and therefore what we consider ‘deep understanding’ or ‘deep thinking.’

2) To overcome it, we must reconsider old lesson and curriculum structures, to carefully introduce greater variation into lessons, which will require us to remove lesson objectives as we know them.


Engelmann, S., & Carnine, D. (1982). Theory of instruction: Principles and applications. New York: Irvington Publishers. accessed 17/04/17

Sweller, J., Van Merrienboer, J. J., & Paas, F. G. (1998). Cognitive architecture and instructional design. Educational psychology review, 10(3), 251-296.

Kahneman, D. (2011). Thinking, fast and slow. Macmillan.

Snider, Vicki E. “A Comparison of Spiral versus Strand Curriculum.” Journal of Direct Instruction 4.1 (2004): 29-39.

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MECE – 7 – Mental Models for Education

Pronounced ‘Mee-See,’ this is another fave from the world of consulting.

When mapping out a curriculum, assessment, taxonomy, decision or driver tree, consider whether its components as you’ve outlined them are Mutually Exclusive, as well as Collectively Exhaustive.

In other words, make sure your categories don’t overlap, and check they cover everything.


I use MindsetKnowledge and Craft as a taxonomy for teacher training.  Each of these breaks down further.

Mindset: Beliefs and behaviours (thanks to Matt Hood for that breakdown suggestion)

Knowledge: Theory, Subject, Pedagogy, Context

Context breaks down further still.

Context: Historical, Contemporary

At each stage, ideally, those boxes wouldn’t overlap, yet together they would incorporate everything a teacher needs to teach effectively.

Teacher Training.JPG


At each stage, ideally, there would be three boxes, and never more than five; remember, this is a tool to aid thinking and help make decisions.  The more complex it becomes, the less it will aid thought.

When thinking about observing lessons, you could try to observe planning, instruction, behaviour, assessment (thanks to Joe Kirby for this example.  To be clear, I’m not speaking about judging lessons through observation, here, merely thinking about things that one can possibly observe.)  Can you really observe planning, though?  It might be inferred, but it cannot be observed.  Alternatively it could be paired back to just Instruction and Behaviour.

What I like about the potential of this as a MECE example is that it could be considered Teacher Behaviour and Pupil Behaviour, since behaviour is ultimately the only thing that can be observed directly.

Teacher Behaviour:  Instruction, Assessment, Management

Pupil Behaviour: Compliance, Self-Direction

Teacher behaviour is therefore broken down into Instruction – things the teacher observably does to communicate ideas to the pupils – Assessment – things the teacher observably does to assess whether or not those ideas were communicated successfully – Management – things the teacher observably does to communicate directions to pupils, and maintain order.  Are there any things a teacher will do that don’t fit in these three boxes?  If so, the three are not Collectively Exhaustive.

Pupil behaviour is interesting.  Can you think of any pupil behaviours that are not compliance with teacher direction, or acted upon of their own direction?  Unless I’ve missed something, it’s a perfect dichotomy, and as an observer you are looking for the extent to which pupils comply with directions from the teacher, and the extent to which pupil actions outside of teacher-direction are good or poor choices.

Classroom Obs.JPG

When thinking about teaching, cognitive science currently considers all knowledge declarative or procedural.  No overlap between them, and if the models are correct, it shouldn’t be possible to think of an example of knowledge that is neither of the two.

I’m not saying that my examples above are perfect, and they might not even be frameworks that you would want to use, but I hope they are at least examples of how this kind of thinking can aid the structuring of complex systems, and in turn aid decision making.

I would recommend reading a few short resources here, here and here for further, and possibly better, examples of MECE from its traditional place in the world of business and management.

If anyone can think of better examples from the world of education, please share!




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Objective Oriented – 6 – Mental Models for Education

I’ve written about this before, so will only touch on it again briefly.

There are two modes of thought when doing something:

  1. Process oriented
  2. Objective oriented

The first asks ‘What must I do?’ then tries to do it.  The second asks ‘What must I achieve?’ and then tries to figure out how best to achieve it.

Most people are naturally process oriented, all the time, and it has two dangers.  The first is that it can make us feel we have ‘achieved’ simply be executing the process, whether results were realised or not.

In government, for example, having ‘distributed leaflets’ or ‘run x number of local information sessions’ might be given as measures of success to justify a programme’s spending, yet, the results (sometimes called outcomes) of those activities is never actually mentioned.  Did they change people’s behaviour in the way you hoped?

The second is that it can make it difficult for us to see ways of achieving better results.  We tend to focus on tweaking our existing process, rather than imagining an alternative way of reaching our objectives.

I give a clear example of this from the world of hand dryers in my original post, linked above.

In teaching, being process oriented would mean you see your job as turning up, doing some stuff that we’ll call teaching, and so long as you do that, you’ve discharged your responsibilities as a teacher.

Being objective oriented would mean you’re always asking yourself ‘But did the children learn what I intended them do?  How do I know?  If not, what am I going to do about that?’

In short, an objective oriented mindset leads naturally to the process of reflection.

What I find ironic in education rhetoric is that by asking teachers to be ‘reflective practitioners,’ we are focusing them on a process, rather than the objective…

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The 80:20 Principle – 5 – Mental Models for Education

In 1896 the Italian economist Vilfredo Pareto noticed that roughly 80% of the land in Italy was owned by 20% of its population.

The Pareto Principle, or more often, the 80:20 principle, suggests that 80% of your sales usually comes from only 20% of your products, services or clients.  It’s the reason a First Class ticket to New York costs £7,000 when an economy class ticket costs only £400.




This one’s brilliant for time management and decision making in general.  It can be combined with Effort:Impact to help us to realise how, actually, all that time we spend trying to make super nice PowerPoints probably falls into the 20% impact for 80% effort category.

The most important use of the 80:20 principle I’ve found in education is curriculum design.  Will Emeny produced this incredible network map of the GCSE maths curriculum, and in doing so he revealed the 20% that underpin the 80% of the curriculum.



  • Multiply and divide whole numbers
  • Add and subtract whole numbers
  • Multiply and divide decimal numbers
  • Add and subtract decimal numbers
  • Understand place value
  • Multiply and divide negative numbers
  • Add and subtract negative numbers
  • Order of operations
  • Round to decimal places
  • Round to significant figures
  • Powers of 10
  • Fraction of an amount
  • Connect between fractions, ratios, decimals and percentages
  • Plot and identify coordinates


If your pupils can’t do those things, they’re not doing much of anything else.  Therefore, focus your Year 7 curriculum here, and make damned sure they all succeed!

Michel Thomas is a master of this.  He pointed out that the 100 highest frequency words in the French language made up 50% of the everything people said in everyday conversation, so with only 100 words you’re half way there!  He made sure to put them front and centre of his language curriculum.

He did the same again by finding rules that opened up the most words possible for English speakers, and introduces them immediately, rules such as all English words ending in ‘-ible’ and ‘-able’ being the same in French, just with a different pronunciation.  Doing so gives you an immediate vocabulary of thousands of French words, so that only 90 minutes in he’s asking you how to say in French ‘What is your opinion of the economic and political situation in France at the moment?’


In physics, chemistry, biology, what are the 20% topics?

What about geography and history?

English?  (Stop saying it’s about nothing more than skills!)

What about music and art?


Figure it out, redesign your curriculum, leverage the power of the 20%.  I put that down as one of the most vital ingredients in the success of Bruno Reddy’s maths curriculum at KSA.


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Cost-Benefit Analysis – 4 – Mental Models for Education

When considering the benefits of your new initiative, think also of the costs.

In business this can be a simple calculation: what are the projected costs?  What is the expected revenue or saving?  Does the revenue / saving exceed the cost?  If so, profit.

It’s rarely ever so simple in education, but the principle is the same.

Towards the end of Elizabeth Green’s book Building a Better Teacher, she tells a story of a teacher who deals with a recalcitrant child by tolerating her slightly abusive tone, and engaging in her game until she ultimately guided the pupil to the point she had wanted all along: asking a question about a film clip they had seen, and with a more responsive tone in her voice.  Green describes the exercise as being like ‘a Tai Chi deflection.’  Her telling of the story is a positive one – this is a teacher to be admired.

Green explicates the benefits, however, without listing the costs.  How much of the other pupils’ time was being wasted while this child played her games?  If only a minute, that’s half an hour of learning time in total (assuming a class of 30).  That number scales up rapidly if they spent two minutes, or three, in this back and forth conflict.  What lessons was this child learning throughout?  That her poor choices and negative actions carry no negative consequences in turn?  That this is positive behaviour, in which she should often engage?

Green outlined the legitimate positive outcomes of the teacher’s decisions, but didn’t pause to list any of the potential costs.  If she had, we could pose the equally legitimate question ‘did the benefits exceed the costs?’

We’re now at a point where we can see that the costs can take on many forms:

  • Money
  • Time
  • Money, in the form of time
  • Opportunity

Cost-Benefit Analysis is like an advanced version of Effort:Impact ratio.  Where the ratio is a very simple heuristic for good decision making, the analysis can be a more formal exercise.


Financial cost-benefit analysis


We can’t necessarily set forth all our costs and benefits quite so quantitatively, but the principle is the same.  All benefits carry associated costs, and we need to be on the look out for them, in whatever form they take, before making our final decisions.



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