Post by Ruth Bull - All opinions are my own.
What does mastery mean?
‘Mastery’ in mathematics has been used as a term for a few years in the UK and the longer it is talked about, the more confused the world of primary maths teaching appears to become. Many schools say they are ‘doing’ mastery. Many teachers focus on ‘mastery’ on Fridays. Some schools have a ‘mastery’ curriculum. Some schools assess with ‘mastery’ statements. So, it all leaves me wondering what does ‘mastery’ mean?
Well, it appears there is no one definition.
Below is my summary of what different stakeholders have said.
Interestingly, the DfE website does not have a definition. So, when looking at a Government viewpoint I was drawn to examine what Nick Gibb (2016) (minister of state for the Department of Education) said when he delivered a speech at the ACME (Advisory Council Maths Education) conference. An account that I have been mulling over since I heard it in July 2016.
He suggested that the international PISA tables indicate that the high performing jurisdictions in south-east Asia have the answers to improving our grades in the UK by using Asia mastery teaching methods. He indicates that countries such as Korea and Singapore and cities such as Hong Kong and Shanghai have low percentages of children who don’t do well and that this is due to their approach to teaching maths. Consequently, he has instigated a programme whereby teachers from the cities of Shanghai and Singapore have come to this country to teach in our schools, and some of our teachers have travelled to see the teaching over there.
He is a big believer that children should be learning the times tables and links this knowledge with fluency: “number knowledge and fluency in written calculation are not the antithesis of problem solving in mathematics. Rather, they are the royal road by which complex mathematical thinking is achieved.” My instinct tells me that the ‘royal road’ may have a thin layer of number facts but the foundations go so much deeper so I looked elsewhere.
Nick’s view is not shared by many of those with an academic interest in teaching mathematics though. I understand that Professor Jo Boaler (2015), a British author and educator currently at Stanford University in California, is not an advocate of times tables learning and would rather children developed an understanding of ‘number sense’ than learned tables facts by rote.
Mr Gibb goes on to say “Maths mastery involves children being taught as a whole class, building depth of understanding of the structure of maths, supported by the use of high-quality textbooks” and has said that £41m will be spent over the next 5 years on supporting more than 8000 primary schools (nearly half of all in England) to adopt a mastery approach. The adoption of this approach is also being encouraged in Secondary schools.
He gives further clues as to what mastery is; indicating that maths lessons in Shanghai are 35 minutes long, additional lessons are given for those pupils needing more support, lessons focus on developing conceptual understanding and then on practice and consolidation of new content, children face the front in rows which leads to more whole-class engagement. No evidence was presented to support this claim. Actually, I think a lot depends on the activity in hand. If we want pupils to discuss their maths then sitting in rows is not helpful. However, if we want them to focus independently on something then sitting in rows might be more helpful. The Cockcroft (1982) report was a big advocate of group work and talk for learning.
He continues to say that when he visited Shanghai classrooms he was “greatly impressed by the emphasis placed on ensuring mathematical procedures and knowledge are underpinned by strong conceptual understanding, often through visual representations. In addition, a great emphasis was placed in these schools on ensuring that pupils use clear and precise mathematical language from an early age to articulate the procedures they perform.” From this can we infer that visual representations and vocabulary are also elements of a mastery approach? If so, I am perhaps warming a little more to his views!
The NCETM (2014) - National Centre for the Excellence of Teaching Mathematics has also issued thoughts on mastery. Later they issued a further document called “The essence of maths teaching for mastery (NCETM, 2016) which summed up the main points. Their view of mastery seems to incorporate many of the ideas that Gibb discusses and indeed the £41m he mentioned is being used to sponsor the maths hub programme which is being coordinated by the NCETM.
They state that maths needs to be for everyone and have adopted a growth mindset in their planning guidance. The idea of ‘growth mindsets’ is something that many have discussed recently. Dweck (2014), a psychologist from Stanford University has carried out research in this area for over 20 years. She states:
A “fixed mindset” assumes that our character, intelligence, and creative ability are static givens which we can’t change in any meaningful way, and success is the affirmation of that inherent intelligence, an assessment of how those givens measure up against an equally fixed standard; striving for success and avoiding failure at all costs become a way of maintaining the sense of being smart or skilled. A “growth mindset,” on the other hand, thrives on challenge and sees failure not as evidence of unintelligence but as a heartening springboard for growth and for stretching our existing abilities. Out of these two mindsets, which we manifest from a very early age, springs a great deal of our behavior, our relationship with success and failure in both professional and personal contexts, and ultimately our capacity for happiness.
More recently Boaler and Dweck (2016) have produced a book on mathematical mindsets, and give detailed views about mindsets in mathematics. They believe children’s experiences and ideas of maths can be transformed through a positive growth mindset. This belief certainly fits in with the mastery approach of everyone doing the same objectives at the same time and that everyone can do maths!
Changing mindset beliefs has a knock-on effect to differentiation. In the past, teachers were encouraged to deliver different learning objectives for different pupils in the same class. For example, by having one set of activities for the low achievers, one set for the average students, and one set for the high achievers. This not only required the teacher to prepare many different resources but required much ‘juggling’ in the lesson in order to get around to all the groups, check everyone was on task, work out who had done what and what their next steps should be. This became an extremely difficult job to do well and one that was nigh on impossible with large class sizes. Those who the maths was thought to be too hard for were given something watered down or much easier or something different altogether. While the high fliers were extended onto something else, possibly from the next year up or beyond.
The NCETM (2016) approach to mastery is that pupils are taught through whole-class interactive teaching, where the focus is on all pupils working together on the same lesson content at the same time. This clearly breaks with the traditional view of differentiation as it was in the past. The whole class approach also fits in with Nick Gibb’s ideas on the use of text books, where everyone can be working on the same exercise or activities at the same time. But what about the slower graspers I hear you ask? Well, any children not understanding a lesson are given additional lessons in order that they may ‘keep up’ with their peers. These interventions should occur swiftly and normally on the same day (as happens in Shanghai and Singapore). In fact, The NCETM (2016) states this too “It is suggested that differentiation is not through subject content but through urgent intervention for those children who are not meeting objectives”. This idea is also adopted by Mathematics Mastery (2014). Differentiation for the rapid graspers is achieved with the focus now on enrichment rather than acceleration. They are encouraged to go deeper with the same learning objectives and concepts rather than be rushed into new subject material.
The NCETM (2016) also agree that times tables should be learnt to automaticity stating:
“Key facts such as multiplication tables and addition facts within 10 are learnt to automaticity to avoid cognitive overload in the working memory and enable pupils to focus on new concepts.”
They go on to discuss ‘variation and intelligent practice’. They recognise that practice is important but that ‘intelligent practice’ develops conceptual understanding while reinforcing procedural fluency. The variation made through exercises is carefully planned to support children to seek and make connections in their learning. Children spend longer on mathematical areas as they develop deep knowledge on the key ideas that are needed to underpin future learning. This last point has implications on medium and long term plans. Topics are not necessarily revisited as they may have been before because longer has been spent on them in the first place! The teacher should spend time on a concept or topic ensuring that all students attain the right level of knowledge, understanding and skill. Have schools picked up this idea? How do we keep all the plates spinning when children don’t return to key ideas?
The National STEM Learning Network
The National STEM Learning Network (2016) (Science, Technology, Engineering and Mathematics), support this idea on mastery when they say:
“Mastery is based on the idea of children not moving on until they are secure in their understanding of a particular concept. The whole class is taught the same thing, at the same time, with children learning at an appropriate level through support and enrichment. Challenge is offered through higher order questioning and activities that develop deeper understanding, problem solving and reasoning skills.”
I can see that children can understand more if they spend longer on something but maths is a web of interconnected ideas and knowledge and children will need to revisit areas in order to build up those connections and that understanding as they move from one area to another. I do think that topics will need to be revisited and built on!
How is Mastery implemented?
The whole issue of planning and curriculum coverage does need addressing. It seems that the mastery approach from the far-east uses a highly planned and thought through journey by the teachers in the classroom, and indeed, they have the time provided to plan, teach and assess thoroughly. I believe I am right in saying that the teachers have over 100 hours of CPD in a year and teach two or three 35-minute maths lessons a day (even as a primary school teacher). This is in stark contrast to what happens in the UK! Teachers here are lucky if they get the designated 10% PPA time and probably get less than 10 quality CPD hours in a year.
Showing the bigger picture
One of the key aspects of mastery identified by the NCETM (2016) is that of ‘Coherence’. Small connected steps – connections not following a narrow pathway but looking to the side and seeing a bigger picture. A kind of enabling children to find the side roads and short cuts between cities (metaphorically speaking)! If the journey through the curriculum is to be coherent, it needs time and quality CPD to ensure that they are able to lay out the ‘royal road’ in such a way that the whole map becomes understood and children can navigate their way through the rich and fascinating territory of mathematics.
STEM (2016) also advocate the use of manipulatives (practical resources used in the classroom) as a means of differentiation. Concrete manipulatives can be used to both support understanding with concepts and procedures as well as deepening and extending thinking with the ‘rapid graspers’. This again is a mind shift for many. In the past, it was only the weaker students on the ‘red table’ who had the resources out while the ‘blue table’ definitely belittled these and saw it as a weakness to need such things to support their understanding!
It seems that Mathematics Mastery (2014) (The organisation developed through the Ark Schools), STEM (2016), NAMA (2015) (National Association of Maths Advisers) and the NCETM (2014) all advocate a CPA approach to mastery. CPA standing for Concrete, Pictorial, Abstract. This isn’t a new concept? This is what Jerome Bruner (1915-2016) was talking about with his research on Enactive, Iconic and Symbolic modes of representation years ago? Also, Caleb Gattegno (1911-1988) popularised the use of coloured number rods created by the primary school teacher Georges Cuisenaire (1891-1975) and incidentally, which are receiving somewhat of a revival under the auspices of ‘mastery’! Zoltan Dienes (1916-2014) invented the base 10 resources to support written calculation methods and place value understanding. His equipment has been in popular use in UK schools for at least the last 30 years and still bears his name. Concrete materials have been around for a while and have always been able to support the understanding of mathematical concepts. It is through the use of mathematical representations that the underlying structure of the maths can be exposed. A view endorsed by NCETM (2014).
What do teachers think of mastery?
It is interesting to note that there is no mention of the word ‘mastery’ in the latest National Curriculum (DfE, 2013) for England. There are three aims for mathematics, namely, fluency, reasoning and problem solving. It is ‘fluency’ to which I now turn my thoughts. So how does this fit in with mastery? The bullet point in the NC document (DfE, 2013) featuring this aim states:
become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
Teachers, that I have met, have wondered if ‘fluency in the fundamentals’ means lots of rote learning on number bonds, tables facts, doubles and halves etc. (with little or no understanding) and there is a danger that up and down the country this is how it may be interpreted. It’s some kind of mechanical repetition that will somehow help students to understand something better by doing lots of it! Personally, I think fluency is wrapped up in ‘number sense’ but that’s another debate.
NAMA (2015, p.6) sum up this concern when they say: “In summary, there is a danger that superficial repetitive practice becomes simply a mechanical exercise, quickly lost in memory and difficult to apply in different contexts.”
The team at Nrich (2015) have also considered what mastery is. They show an intertwined rope image first shown by Kilpatrick et al (2001) that demonstrates what they think is needed to nurture our young mathematicians. Image shown below.
They feel that mastery addresses procedural fluency and conceptual understanding well but does not address all the attributes needed to nurture young mathematicians.
They don’t discuss any of the issues around what mastery might look like in terms of CPA, differentiation and planning.
Finally, when looking at problem solving (one of the main aims of the NC) – this fits in well with the mastery approach to ‘mathematical thinking’ as highlighted by the NCETM (2016) and Maths Mastery (2014). Both organisations list this as one of the key things to be doing and it should tie in with reasoning and fluency too in my opinion. STEM (2016) say “Challenge is offered through higher order questioning and activities that develop deeper understanding, problem solving and reasoning skills”.
Certainly, this is what maths should be about. If we can’t educate children to be ‘problem solvers’ then we have missed the whole point.
One last point. Assessment, mastery style – it’s not a checklist. You can’t tick it off! Nor can one say their children have done mastery like it’s some kind of outcome. How can we say someone has mastered something? Do we ever stop learning? It’s a bit like saying 100 is a big number, then we find out about a 1000 or a million. There is always something else to know or learn on an area. Isn’t there?
In conclusion, I see ‘mastery’ as an approach and as a set of beliefs and I don’t think it is very different to what good teaching has always been about. I don’t think it is an ‘outcome’ or something that can be ticked off on a tracking system. I think it is about a learning journey through maths that is designed to encourage children to make connections in their maths. It’s about using a CPA approach to draw out underlying mathematical structures and to help learn concepts. I think problem solving is at the heart of it and that children should be encouraged to address this in a positive way. I think as teachers, we should be modelling our thoughts and approaches to problems, including our trips up the dead ends! All too often we give problems that are quickly solved and we expect books to look neat and tidy with a clear answer at the bottom. Real problems are not necessarily like that. I welcome the mastery approach of single learning objectives and whole class teaching and the changes this brings to how we can differentiate.
- Boaler J., (2015) accessed online at: https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/2015/03/FluencyWithoutFear-2015.pdf [Accessed 13th February 2017]
- Boaler J., & Dweck, C., (2016) Mathematical Mindsets: unleashing students’ potential through creative math, inspiring messages, and innovative teaching, Jossey Bass.
- Cockcroft, W.H., (1982) Mathematics Counts, accessed online at: http://www.school-maths.info/1982_Mathematics_Counts.pdf [Accessed 15th February 2017]
- Department for Education (DfE). (2013a). National Curriculum in England: Framework Document.
- London: Department for Education.
- Dweck, C., (2014) accessed online at: https://www.brainpickings.org/2014/01/29/carol-dweck-mindset/ [Accessed 14th February 2017]
- Gibb, Nick, Department for Education (DFE), corp creator. (2016) Speech : Nick Gibb : building a renaissance in mathematics teaching : 12 July 2016, ACME conference, The Royal Society, London accessed online at http://dera.ioe.ac.uk/26970/ [Accessed on13th February 2017]
- Kilpatrick, J. Swafford, J. & Findell, B.(eds.) (2001). Adding it up: Helping children learn mathematics.
- Mathematics Learning Study Committee: National Research Council.
- Mathematics Mastery (2014) accessed online at: https://www.mathematicsmastery.org/our-approach/ [Accessed 13th February 2017]
- NAMA (2015) accessed online at: http://www.nam0Myths%20about%20Mathematics%20Mastery.pdf [Accessed 15th February 2017]
- NCETM (2014). Developing Mastery in Mathematics. [Online] Available from: https://www.ncetm.org.uk/resources/45776 [Accessed: 13th February 2017]
- NCETM (2016) accessed online at: https://www.ncetm.org.uk/files/37086535/The+Essence+of+Maths+Teaching+for+Mastery+june+2016.pdf [Accessed 14th February 2017]
- Nrich (2015) accessed online at: https://nrich.maths.org/11796 [Accessed 15th February 2017]
- STEM (2016) accessed online at: https://www.stem.org.uk/blog/mastery-primary-mathematics [Accessed 15th February 2017]