Mastery does NOT mean full understanding

“For now we see in a mirror, darkly; but then face to face: now I know in part; but then shall I know fully even as also I was fully known” 1 Corinthians 13 v12.

‘Mastery’ means ‘full understanding’ according to many teachers on twitter. So ‘mastery teaching’ means not moving on with your course until something is fully understood. I don’t think teachers really think this because there is a one insurmountable problem with this definition of mastery – there is no such thing as full understanding. For example:

  • When a KS1 child is first taught ‘place value’ is it conceivable that they can fully understand the notion, with all its implications? Surely many GCSE students could do with understanding place value better than they do?
  • My year 10 history class use the word ‘dictatorship’ with some confidence in their writing suggesting they understand it but sometimes they do use the term incorrectly so do they understand the full implications of the term? I know they don’t because I have a better understanding of the term than they do. Do I understand the full implications of the term dictatorship? I know I don’t because the historian Richard Evans definitely understands it better than me.
  • My eight year old son has started reading Harry Potter books by himself. Does he understand them? Well I don’t suppose he realises (as J K Rowling must have appreciated) that the Hogwarts house elves illustrate the Marxist notion of false consciousness. I don’t even think he gets the same depth of meaning from these books as his thirteen year old sister. So when will he be able to ‘master’ Harry Potter? Should he wait to read them until he is able to gain an appreciation of Marxist theory or just until he is mature enough to understand Harry’s teen romances?

In reality of course teachers, as professionals, don’t hang around waiting for FULL understanding – that would be ridiculous. They actually make sensible decisions about the ‘degree’ of understanding necessary for a child at that stage with the curricular content they are learning. The word, ‘mastery’ can’t tell us a thing about what this sensible degree of understanding might be.

Unfortunately the mistaken notion of ‘full understanding’ is not harmless in practice. It can mean teachers do hang around for too long focusing counter-productively on ever greater understanding. A maths teacher may be convinced that a KS1 child must fully understand place value when the notion has been taught at a basic level. They may introduce word problems to check for mastery or ‘full understanding’ of place value. In their pursuit of ‘full understanding’ they fail to consider:

  1. Ability to use learning in new contexts (like word problems in maths or knowledge in history sourcework or applied GCSE science questions) tends to lag behind initial learning because newly learnt knowledge is what is called ‘inflexible’. To overcome this inflexibility you need to accumulate a greater store of related knowledge, facts and examples.
  2. In the case of reading, holding children back so they can ‘fully understand’ what they read, can mean they lack exposure to the very new words and ideas that will allow greater understanding to develop.
  3. As Willingham explains, knowing more facts makes many cognitive functions such as comprehension and problem solving operate more efficiently. Therefore a focus on memory (really knowing what is taught long term) as well as initial understanding is important. This means better understanding often develops after greater FLUENCY OF KNOWLEDGE has been achieved so, for example, lots of practice gaining confidence and really knowing a mathematical method can open up the possibility of further understanding of related concepts. Knowing more about the causes of World War One will make it more possible to demonstrate understanding in an essay.

I like to think of understanding and fluency of knowledge as the partners in a traditional dance. Sometimes they work in unison:

Netherfield Ball 4 (2)

And sometimes they work apart, one going before the other, like dancers executing moves that do not involve their partner.

netherfield ball 1

This means, dare I say it, sometimes it makes sense to teach knowledge and ensure it is remembered even though it means understanding lags behind. It is the teacher that needs to decide whether greater fluency of knowledge or greater understanding is more necessary at any given point. When making this decision perhaps we should bear in mind that in modern education the trend has been towards overemphasising initial understanding at the expense of necessary fluency of knowledge through ensuring that what is taught has been remembered confidently long term.

Where does this leave the word ‘mastery’? We’ve already established that mastery is not a principle we can use to judge the degree of detail in which students must grasp curricular content. Mastery can, however, describe how well children have grasped or can perform whatever the teacher has considered that they need to know or be able to do at that given point  whether that is fluency of knowledge or understanding. When used in this sense the term mastery is useful. The confusion occurs because teachers think about ‘mastery’ in curricular rather than pedagogical terms:

Curricular decision: What should I teach? I should teach this concept fully

Pedagogical decision: When should I move on? When they understand and have committed to long term memory what I have decided they need to know.

The latter pedagogical goal is a useful way to think about mastery. The former curricular goal is actually impossible (unless, perhaps you are in heaven with God and the angels…)



11 thoughts on “Mastery does NOT mean full understanding

  1. There is a real issue with terms being ill-defined and the world of teaching seems particularly beset by this problem. It means that things are open to interpretation and spin, according to political or commercial interests etc. I agree with your interpretation but perhaps the term is a red-herring anyway. Have I mastered it? I don’t know. Do I have a pretty good understanding and facility so that I’m ready to move on? Possibly.

    1. I do think the term mastery is useful in the sense it was used by those first promoting it.
      There is a tendency within teaching to adopt developmentalism assumptions. You expose children to content and they learn it when they are ‘ready’. I think of this as the ‘mud on the wall’ approach. You fling the mud and just hope some sticks. Mastery is the correct word to describe the educational alternative to developmentalism assumptions. With a mastery approach you identify what the children need to know next and you ensure ALL are rock solid before moving on.

      1. Nice clarification. I like to think of it as what is essential that every student knows/can do each lesson and then not move on until this is mastered. It definitely should be a part of lesson planning and probably added to schemes of work.

  2. The “Number Line” is a classic here – a useful tool, but with a multi-layered understanding. It first gets filled with positive integers and counting forwards and backwards and then fractions and decimals and negative numbers. At some stage, if a student takes enough maths the Real Numbers will be introduced, but probably made fully rigorous only on a degree course. Then if you wish, you can add a point at infinity and wrap it around a circle. or follow JH Conway “On Numbers and Games” and add innumerably more numbers including infinitesimals. The line becomes part of the complex plane too. And there is yet more to be explored through images to the number line – curves (and space-filling curves). There are what I would call “layers of understanding” here.

    Or take the forward roll – one of the olympic gymnasts came out of a tumbling routine into a perfect forward roll. That is a level of “mastery” in terms of control and precision of movement beyond general primary education.

    Which doesn’t mean that we avoid accuracy in mathematical expression or precision in mathematical thinking, or that we don’t develop precision and control of movement in PE, but rather that we have a clear sense of what that looks like and what is attainable in each layer or level of progression. One danger with “mastery” is a concept is that it suggests that all the work is done – and as you suggest in the post there is much to do to extend, contextualise and consolidate learning as the curriculum develops. If a gymnast had “mastered” a forward roll in primary school and didn’t develop it further (including as the body grows and changes), it would not look good in an olympic routine!

  3. To me, the word ‘mastery’ is a red herring, because if we ever think we have ‘mastered’ anything, then we are fools. The sign to me of someone who is good at what they do is that they think the opposite – that they are nowhere near achieving what they hope they will eventually achieve. It strikes me that it’s the antithesis of high expectations to tell kids they have ‘mastery’ of anything.

    I guess there’s a bit of a habit in education of using what we think are important sounding words that end up being mangled, misused and misunderstood until they barely mean anything at all. I think most of us know what mastery means outside of school, and maybe this is where the confusion has arisen, because it’s not being used accurately in the educational context? Perhaps the term ‘grasped’ might work better? (Plus it would get us away from the awful 50 shades of grey undertones of ‘mastering’ anything or anyone).

  4. I’m very relieved my daughter finished primary before mastery maths arrived, where to add to the fun there is ‘mastery’ and ‘mastery with greater depth’.

    I didn’t pay close attention, but the latter looked like an essentially post-hoc addition for high-attainers. Mastery-maths may or may not raise the low-end attainment, but that’s clearly the focus. Despite a little campaign to convince us otherwise, I think it does have the high-end treading water. Worse, attainment gaps between disadvantaged and advantaged are passé, it’s unconditional low-high attainment gaps that should be closed now and it obviously helps if you cap the top. The inconsistency between this and wittering on about every child’s ‘full potential’, now flirting with new grammars, doesn’t appear to trouble anyone which is grimly fascinating.

    Grasped might be better, but short of denying human nature it leads directly to things like ‘fast graspers’ and I can’t help thinking of people at some boxing-day sale.

      1. Every summer we get prodigy stories in the press and the most common one seems to be a primary age child who has achieved a shiny GCSE A* in maths. At secondary level it isn’t, or wasn’t (the new league table first entry rule), unusual to trip over schools doing early-entry, or extra GCSEs in stats/further/additional and sometimes AS level. Maths doesn’t have so much dependency on life-experience (broad knowledge, concepts, vocabulary, imagery) as several other subjects and it’s *the* subject that isn’t easily divorced from aptitude and pace.

        The curriculum has ‘depth before pace’, but NCTEM and others have strongly promoted ‘depth *not* pace’ in their influential version of mastery maths. Around this time last year some parents with brighter children in primary schools began to howl because the depth their child was experiencing was shallow. Mark McCourt’s recent #MasteryFail piece covers almost everything I’ve seen in their stories and although I’m inclined to believe Mark’s version of mastery is much better because they explicitly acknowledge the existence of ‘really bright kids’ in their ‘We don’t do differentiation now, we do mastery instead’ item, I still struggle to see where the depth can come from in relatively basic primary level maths.

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