1. Know your learners
Get to know how they think about maths, how they learn, what they find easy and what they find difficult. Numeracy is an important tool in modern life and the modern workplace, but being adequately numerate requires a concept of numbers, let’s call this numbersense, a knowledge of learning, the ability to think mathematically, and the skill of wading through a whole raft of cultural misconceptions, memes and expectations without losing all sense of confidence and efficacy.
Assess your learner’s basic numeracy skills, their comfort with the rules of arithmetic, and note any misconceptions that need acting on immediately. No doubt you will all have learnt not to expect complete mastery as learners move to whatever level you have been working on, so get to grips with their errors and misunderstandings.
It’s just as important to know how your learners feel about mathematics and learning in general. Do they have a growth mindset? Do they have a particular mathematical learning style? Do they see errors as disasters or places to grow from? Do they have support at home? Are they anxious about maths? How do they do in other subjects?
Professor Steve Chinn’s excellent book ‘More Trouble with Maths’ (NASEN, 2016) has a number of useful resources that will help you to get to know your learners.
The ‘Myself as a Learner Scale’ (Imaginative Minds, 2014) by the late Professor Robert Burden is a fantastic tool for analysing self-perception of learning and can easily be administered in a group really quickly.
2. Be empathetic and sympathetic
Now you are aware of the learning strengths and difficulties of your learners, act on this to provide a learning environment that avoids creating unnecessary discomfort. In ‘The Assessment of Self-efficacy’ (Equals vol. 20.1) I introduced some research that identified aspects of the maths classroom that students lacked the efficacy to operate successfully in, things like explaining a solution to a problem in front of the group and having to answer quickly. Whilst both attributes are somewhat useful, they are also somewhat stressful. There needs to be a balance between understanding the learning value of challenging situations and the impact on a student’s engagement and enjoyment in learning.
Every learner is different, and, whilst it is nigh on impossible to accommodate every individual difference, we should have great regard to the types of difference and the impact this may have on their ability to do mathematics. To give an example, people process information at different speeds, but this does not automatically impair their mathematical ability. Indeed, from my own experience, some of the most accomplished mathematicians are very slow at processing information. With this in mind – how important is the ‘need for speed’? Surely we are after efficient mathematicians who work with confidence and accuracy – highlighting speed of processing issues doesn’t help anyone and can create serious anxiety, even amongst the most capable.
It is also worth remembering the frighteningly common cultural meme – “I was doing OK at Maths and then I had a teacher that hated me”. The thing is, as a bunch of people, maths teachers don’t portray an all pervading sense of dudgeon for their learners, and, whilst we may get frustrated with some, shouldn’t really dislike anyone. The problem is this, our learners are young and trying to find their way in the world – they are sensitive souls – and therefore, what we consider to be gentle encouragement to engage is seen as something different for learners who find the subject to be difficult and teachers to be lacking in empathy. If we remember this, we may be able to head off such disillusionment before it becomes disengagement.
Read the case studies on Steve Chinn’s ‘The Fear of Maths. Sum Hope3’ (2012, Souvenir Press) and ‘It Just Doesn’t Add Up. Explaining Dyscalculia and Overcoming Number Problems for Children and Adults’ by Paul Moorcraft (2015, Tarquin Publications) for an insider’s viewpoint.
3. Use a range of approaches to learning.
In particular, consider the steps that are gone through when we learn new maths ideas. Bryant, et al. (2015) identify the ‘concrete – semi-concrete – abstract’ (CSA) teaching sequence, whilst Professor Mahesh C. Sharma describes the six levels of knowledge required to understand a new mathematical idea – intuitive; concrete; pictorial; abstract; application; and, communication.
The central theme running through both sequences is the movement from concrete manipulatives – for example, Dienes Blocks, Cuisenaire Rods and fraction tiles – through semi-concrete, pictorial, representations to abstract methods which use symbols.
Sharma further develops this by suggesting that the ideas learnt need to be applied in a range of situations, and then reinforced by communicating the idea to others, be that the group, an individual of the teacher.
Approaches that allow for a visualisation of how numbers work are used throughout the NCETM Calculation Guidance for Primary Schools (2015) and are common in Singapore Maths.
The NCETM Guidance is available here: https://www.ncetm.org.uk/public/files/25120980/NCETM+Calculation+Guidance+October+2015.pdf
Read about Professor Sharma’s ‘Six Levels of Knowledge’ Model in Judy Hornigold’s ‘Dyscalculia Pocketbook’ (2015, Teacher Pocketbooks series).
The chapter ‘Learning Disabilities. Mathematics Characteristics and Instructional Exemplars’ by Diane Pedrotty Bryant, Bryan Bryant, Mikyung Shin and Kathleen Hughes Pfannenstiel is taken from ‘The Routledge International Handbook of Dyscalculia and Mathematical Learning Difficulties’ edited by Steve Chinn (2015, Routledge). This book contains a number of other useful chapters on evidence based classroom practice.
Professor Sharma’s ‘Six Levels of Knowledge’ model, above, begins its sequence with intuition. Each new fact is linked to something the learner already knows. Ideally, this is linked to something that the learner can do with an element of automaticity. At its simplest level, mastery is being able to undertake a task with fluency and understanding, so that it can be completed with little cognitive load. How often do we feel that our learners have achieved mastery before we move on to the next topic?
Salman Kahn, the founder of the Kahn Academy, makes an intriguing analogy in his TED Talk ‘Let’s teach for mastery – not tests scores’ when talking about knowledge gaps that reappear later in the curriculum. Imagine building a house and saying you have two weeks to build the foundations, and then something happens, it rains for a week, or the apprentice mixes the concrete wrong, so that the foundations are only 70% complete – you wouldn’t just move on to building the first floor, and you certainly wouldn’t get the electricians in before you have completed the roof. Unfortunately, we often feel compelled to move on before we have built the solid foundations, which will only lead to struggling learners in the future.
5. Discussion – at every opportunity
There are three discussions that should be encouraged in the classroom; discussion for learning, where learners are engaged on whole-class and group tasks together and discuss their approach to problem solving, address gaps in knowledge and create a collaborative and supportive environment. Jennie Pennant provides a useful resource on the NRICH website to support this (https://nrich.maths.org/10341); Discussion as feedback, where the teacher is able to provide constructive and challenging support to help learners develop understanding. This model may draw on the master and apprentice relationship, where the learner is guided towards their own mastery through discussion and a feeling of belonging to a community of practice. ‘Situated Learning’ by Jean Lave and Etienne Wenger (1991, Cambridge University Press) is a benchmark text. In many ways, if we are empathetic teachers, we already employ this approach; And, finally, discussion for metacognitive reflection, where we encourage our learners to adopt a growth mindset and view errors as learning opportunities, or just to think ‘if I did that wrong this time, what do I need to do to avoid making the same mistake’. Read John Hattie and Gregory Yates’ ‘Visible Learning and the Science of How We Learn’ for some evidence led analysis.
6. Have Clearly defined classroom rules that are on display and are adhered to.
Everybody loves it when they know where they stand. If both yourself and the learners know what is expected in the classroom, you will create a positive environment that is disciplined rather than an unruly one requiring disciplinary action – although we have all had that wet Friday in February!
If we create a positive environment in which people believe they can learn without ridicule, where failure is renamed practice, and where everyone is treated with respect and have clear rules that are targeted at creating this environment, then we all begin to expect learning to happen. ‘The Essence of Maths Teaching for Mastery’ from the NCETM sets out some clear expectations that should be made available to learners. ‘Engaging Learners’ by Andy Griffith and Mark Burns (2012, Osiris) has an excellent chapter on extrinsic and intrinsic motivation.
7. Be fun and energetic and passionate.
Because you love mathematics, it is the most incredible subject, it surrounds us in our everyday lives and you want everyone to be able to access the world of numbers and to learn how they can explain the world around them.
Try reading ‘Mathematical mindsets’ by Jo Boaler (2016, Jossey-Bass) as inspiration.