Mastery Learning in secondary maths- re-imagining everything from first principles
Every now and again you have to go back to first principles and look at problems, systems, technologies from a blank canvas perspective. Life is full of trade-offs where you need to make a decision about where you lie on that particular spectrum. You can choose to pay more or less for posting a letter but the trade-off for paying less is second, rather than first class delivery. Trade-offs don’t go away by optimising where you choose to lie on the spectrum. Choosing whether to post a letter first or second class post doesn’t take away the trade-off that you get what you pay for. To get high speed posting of the same letter you need a paradigm shift. Most often, paradigm shifts come from new technologies. Send that same letter fast via email and it won’t cost you a penny. The technological paradigm shift has eliminated the speed-cost trade-off. I use the word technology in the loosest sense, meaning new systems, ways of thinking, ways of working, design, engineering, as well as information and communication technology. It is inventions, whether tangible or intellectual, that assign trade-offs to history.
Inventors re-imagine things by exploiting new knowledge or technologies. The knowledge gained from scientific discoveries presents opportunities for inventors to explore. Understanding the atom lead to nuclear power. The real trade-off busting inventions always come from going back to first principles, eliminating assumptions that are chained to history and that gather authority from the length of time they have been accepted as conventional wisdom. Elon Musk, the American business magnate, engineer and inventor knows this. In founding Paypal, Tesla Motors and then Solar City, he has started three globally dominant companies each from scratch that has completely revolutionised three different industries (online payments, electric cars and solar power). His latest venture, SpaceX is a solid bet for being the replacement to NASA’s flailing space-program. In a TED interview, Musk talks about how he went back to first principles when thinking about how to design a rocket.
By looking at the relative cost of parts of a rocket he realised that if a rocket could be made reusable, the cost of putting a satellite into orbit could be reduced by one hundred times. In launching a rocket, the fuel cost is just 0.3%, and so why have single-use rockets that throw away nearly all of your investment each time? SpaceX are busy creating reusable rockets that will soon make redundant the single-use rockets that have been conventional wisdom for decades. We will look back and wonder why single-use rockets were ever the norm.
Where do his ideas come from? As Musk puts it:
A good way of thinking is physics’, first principles reasoning. What I mean by that is boil things down to their fundamental truths and reason up from there, as opposed to reasoning by analogy. Through most of our life we get through by reasoning by analogy which essentially means copying what other people do with slight variations. You have to do that or you wouldn’t be able to get through the day. When you want to do something new you have to apply the physics approach. Physics allows you to discover new things like quantum mechanics that are counter-intuitive. Also, really pay attention to negative feedback and solicit it, especially from friends. This may sound like simple advice, but hardly anyone does that.
How has a typical maths classroom changed in the last hundred years? Pupils sat behind desks. Board at the front. Teacher being didactic. Pupils practising a series of questions on a skill just taught. All the class working on the same topic. Setting pupils by attainment. Homework. If you compared typical contemporary classrooms those from a century past you would find more similarities than differences. The basic structure and model is the same. Watch me, copy, practise.
That’s not to say there has been no progress. The model is more refined than ever. AFL, IWBs, peer-assessment, card sorts, thinking time, think-pair-share, reducing teacher talk, PELTS, pupil plenaries and endless other strategies have optimised the direct instruction approach and kids get a better education than ever from committed inspiring teachers.
Nonetheless, what if we went back to first principles? What would remain? What would be different?
We could debate endlessly what should be on the list of what a perfect maths education system would look like, but I think there would be general consensus on these things:
- Students learn about, understand deeply and can apply many mathematical concepts spanning a broad range of topics across the subject, particularly those that will empower them in making key decisions in their lives after school.
- Students should feel they are good at maths and enjoy learning it.
- Students should be driven by intrinsic motivation based on their beliefs about themselves as learners rather than by extrinsic motivation strategies.
- Students should experience the awe and wonder moments, the beauty and mystery and the fascinating history of the subject.
- Student should be nurtured to become life-long, independent learners.
- Students, parents and teachers should be held equally accountable for the student’s progress.
- The learning environment should be such that every student can progress as fast as possible.
- In measuring progress, developing depth of knowledge should be held in equal regard to breadth of knowledge.
- Students should develop excellent social skills, and show respectful behaviours towards all others.
- The quality of teaching that pupils receive should be consistently excellent across the department/ school/ country.
If we believe in these things then the system must have them at its core. They must be the bones of the learning environment, not jewellery.
Think about what the average maths classroom looks like and ask yourself if the routines, structures, rules and ways of working seem logical practical implementations of these ideals. I would argue in the quest for ‘rapid’ progress breadth is nearly always prioritised over depth. Many students do enjoy their learning, but many do not. In most cases this is a vicious circle of failure to learn, reduction in self esteem, destructive coping behaviours, failure to learn etc. The balance of accountability of pupils’ learning has never laid closer to the feet of teachers. Awe and wonder moments do happen, but only when enough planning time is available amongst the daily bureaucracy. Independent learning is encouraged but ultimately flawed if the accountability for the progress is not shared with the student (detentions don’t make pupils accountable for their learning). In a well-pitched lesson half the class could go faster, half need to go slower (ignoring differentiation, but I’ve seen very few teachers do this really well). Students receive endless extrinsic motivation carrots. Good qualifications guaranteeing a good job isn’t true anymore and the kids know it.
Real-world constraints dampen the ideal. Pupils that cannot behave. Learning difficulties. Red tape. Lazy pupils. The tail wags the dog and the entire system of teacher-only accountability for exam results creates a system whereby teaching pupils to answer exam-style questions is often never far from your mind. No re-imagining of the system is going to change any of these things. There will always be students that don’t care, won’t behave and are lazy. No system is going to be a catch-all. The only question is can we think of one that would be significantly better than the current one? Not a catch-all, but a catch-more. It will never live up to the ideals above, but if the day-to-day reality is closer to them it would be worthwhile.
I need your help. The proposal below needs a critique. Is this idea a good one? What have I not thought of? What would the downsides be? Would the day-to-day reality be better learning for pupils? Could/ would it work?
Most schools operate a spiral-based curriculum. Each topic is taught twice over the five years (KS3 and then again and built off at KS4). How many times do you find yourself completely reteaching topics? No system will avoid pupils needing to revise and recap content if time has elapsed from went it was learned. However, if the original learning was deep and based on understanding, rather than recall the previous learning should come back much faster when it comes to recap time. If we ensure pupils master a topic before moving on there would be so many potential benefits. By master, I mean really understand. Saying they get it because they can copy a series of steps you have shown them and then do it with practise questions where the only difference is the numbers is not mastery. Using Bloom’s Taxonomy, understanding is a spectrum and recall is the lowest level of understanding there is. We need to have higher expectations of what it means for a pupil to master a topic.
Until they can describe links between the current topics and other areas, apply their learning to problems in different contexts, analyse what mistakes could be made on particular calculations, evaluate different calculation methods for efficiency and create their own lessons on topics, they haven’t yet mastered it. Let’s base the system on mastery. Let’s not move on until they’ve mastered their learning on topics.
Create a scheme of work with learning units based on objectives at similar levels on a particular topic. Ensure progression is managed. The current levels system could be used as a guide. Do away with exercise books and replace them with portfolios. When students start a topic give them a list of the learning objectives they need to master to complete the learning unit and the success criteria in the form of which exercises/ activities they need to complete. You could also give them info about what relevance this maths has to the real world. The idea is that pupils must complete the set work for you to ‘sign them off’ on a topic. Managing standards, like presentation of question/ working/ answer etc would be straightforward as you are the gatekeeper in signing them off. Ensuring all pupils had appropriate notes, completed exercises and master activities would be possible in a manageable way.
Pupils self-mark and you do comment-only formative feedback.
Take your A game to the planning of the set work. Ensure it’s rich in historical links, engaging discoveries, a mixture of teaching strategies etc. By planning this in advance, the day-to-day red tape that reduces your planning time wouldn’t reduce the quality of what is delivered in class. You could ensure consistency of high quality learning activities and resources across your department, ensuring all the main misconceptions are addressed in every classroom through the high-quality prescribed exercises.
How are they going to learn? You would need to divide up your lessons between ‘teach’ and ‘self-study’. You would still teach either the whole class if appropriate, or teach smaller groups if some pupils were on further forward/ behind. There needs to be resources for independent learning if some pupils have moved onto the next learning unit. We live in a world where this is possible now through things like MathsWatch. For self-driven pupils it is amazing what they can teach themselves from ‘old’ SMILE sheets. They are a masterclass in discovery-based learning. Realistically this needs to be delivered electronically or it would become too paper-heavy so a class set of tablets, laptops etc would be needed. What to teach, when, to whom would be your decision as a teacher based on the AFL you’d be getting from where the pupils are with the work. AFL would quite simply have to be done or you’d have no idea what to teach. That’s a good thing.
What if they can already do the topic? Have an entry assessment for each learning unit. If they can meet some of the learning objectives for the unit already, sign them off on those ones straight away so they are only working on the gaps. If they can do the whole topic, great, move on! This would ensure no gaps in knowledge. If they’ve worked their way through the learning units and demonstrated their knowledge on each one, even if through just a single entry assessment for each topic, you could be sure of their foundations.
What about when they forget things? You would need to build some form of revision activity into your learning units to recap prior knowledge. State which units they need to have another read through if they need to look back deeper.
How do you assess them? If they have got to a particular part of the scheme of work and have done so by demonstrating real mastery on every learning unit up to that point, they don’t need summative testing to check they’ve learned it. Every child has an A* (or 9!) grade from day 1. All they need to do to keep it is to be at a particular point in the scheme of work at intervals throughout their 5 year school career. For example by the end of year 7 they should be 1/5 of the way through all the content up to the A* grade. If they are only 1/5 of the way through all the work to get to a C grade, then tell them so. By looking at their rate of progress through the scheme you could forecast their final grade based on extrapolation. “If you carry on making the same rate of progress as you have so far you would end up with…” Or, “you’re in front/ behind where you need to be to end up at an A.” You would still need lots of time for revision in year 11. This would solve the issue of mathematically able pupils who are just bad at exams ending up in lower attaining classes, studying maths they can do already. We/ they need to sort their exam technique out but that doesn’t mean they shouldn’t be taught the higher level topics if they can handle it.
The whole system would get the kids to understand that to have learned something you need to be able to do a lot more that just recall it. Google makes fast-knowledge available these days like some American corporations make fast food. Fast knowledge isn’t learning though, just as fast food isn’t food. Pupils need to know this and it is built into the core of a mastery-based system.
The system would share responsibility and accountability for the pupils’ learning between the teacher and the child. If they won’t put the effort in they simply won’t progress. There are kids in the current system like this. What is different with this proposal is that the fact they are not progressing because of their lack of effort is much more obvious to them. For the driven pupils they can progress truly at the speed they wish. It is a meritocracy based on effort and attitude rather than natural-born intelligence. At one (or more) point(s) throughout the year re-set the classes based on where the pupils are.
I think it meets the ten ideals if it could be made to work in a practical way. Independent learning, higher order thinking, balanced accountability, well planned and resourced learning is embedded in the core of the system rather than as a bolt-on.
What do you think? Leave your thoughts in this Padlet:
I’ll leave you with a lovely video of the great physicist Richard Feynman talking about the difference between knowing the name of something and knowing something.