The idea of allowing children to become open-minded and think for themselves has always fascinated me.
I think that this stems from my own experience as a child of being taught by teachers who generally imparted knowledge that was predominantly ‘fact based’ and had to be learned and then regurgitated in ‘parrot fashion’, which took lots of practise and exercises in memory recall.
When I trained to be a teacher (later in life), this idea of the educator being purely the ‘font of all knowledge’ was long gone. I had to considerably adapt my previous understanding of how teachers teach and ultimately its impact on what constitutes effective teaching and learning.
Teaching assistants are, of course, an integral part of that outcome and the role provides opportunities to work with small groups of children or in one-one scenarios. They are in the privileged position of being able to listen, to question, to intervene and to make a difference to children’s learning right there in the moment.
Working with small groups of children allows time to be spent on providing focussed attention that may not be available in whole class situations. One useful strategy that can be employed is outlined in an example shared with me by a colleague, which is explained specifically in the work of Bakker, A., Smit, J. & Wegerif, R. (2015).
This story illustrates how a father came to realise his child had a different way of understanding a mathematics problem to his teacher:
‘A boy came home from school and showed his father a mathematics problem that had been marked as wrong. The boy was confused as he thought that his answer was right. The question was a hexagon with a question mark (Fig.1). The answer was the angle. The boy had put 90° and did not know why this was wrong.
His father explained that it was a hexagon so each of the triangles was equilateral and all the internal angles of each triangle were 60° so the answer had to be 60°. He did so in a way that tried to ‘scaffold’ the information, breaking it down into parts and checking for understanding at each stage but it was clear that the child already knew the argument about hexagons and equilaterals but still remained baffled.
It was clear that they did not understand each other.
Eventually, after some false starts, the boy made it clear that for him, the shape was not a hexagon but a cube (Fig. 2). Squinting his eyes and looking again as the child had looked, the father came to see that the image could, in fact, also be seen as a cube. As a cube, the boy’s answer of 90° was correct.’
You may need to look at the shapes several times and squinting definitely helped for me when I first saw it. Think of one shape as two-dimensional (2D) and a hexagon and the other as three-dimensional (3D) and a cube. Adding shading to one surface might help:
The father had learnt something by using ‘Dialogic Discourse’. To allow the children to become open-minded and more independent and less reliant on the teacher, we have to teach in a way that allows for situations to be seen differently.
What’s the same? What’s different?
In the story, the father was teaching for creativity by listening, by pausing, by showing respect and by allowing himself to be led by the child and to learn from the child. Dialogic Education is not just one-way but two-way. The aim is not just to reach the correct answer but also to be able to see things from multiple perspectives.
Here, the child learned to try to work out how the teacher was seeing things and what answer he was supposed to be giving while also maintaining the ability to see things in his own, different way. The father learned that what he had assumed was a simple problem, could be seen in more than one way.
Dialogic teaching then, is about providing open-ended learning dialogues. In this case, it involved asking appropriate questions without ‘closed’ answers, showing respect for other views, however wrong they initially appear, and being open to taking on new perspectives.
The general point here is that ‘scaffolding’ (supporting the child) into a correct answer assumes awareness of a cultural and historical context that make it the correct answer. That is to say, do not assume anything from the outset and begin from a ‘low’ starting point, which could be as simple as asking open questions such as ‘tell me what you think’ or ‘what can you see?’
If exploring this with children, it could also be the case that you show both of the pictures in Fig.2 and ask the question ‘what is the same and what is different?’ These types of questions can be used by adults supporting children’s learning to explore children’s ideas first, rather than focusing on solely arriving at the ‘right’ answer.
Identifying gaps and misconceptions ‘in the moment’
Another big advantage of working in this way is that it can also reveal any misconceptions or errors early on which can then be addressed in the moment or through subsequent teaching.
As an example, I recently worked with a group of teaching assistants and a small group of children in Key Stage One. We looked at the following question but just showed the children the first statement and the picture and asked them for their ideas about what it meant. We also had counters as ‘sweets’. The rationale for this was to ‘spotlight’ the concept of ‘more than’ to ascertain if the children understood it. We did not explain or guide or instruct.
From this ‘low’ starting point, it became clear that the children understood ‘more than’ but were confused by the phrase ‘no more than’. To that end, we then taught the concept of ‘no more than’ and allowed the children to explore it using the concrete resources (the counters) and number lines. We also made a fluency slide for the children to revisit regularly, with different amounts displayed each time.
We then took each part of the question separately as another focus or ‘spotlight’ e.g. ‘she counts her sweets in groups of two’ and explored each concept before showing the whole question for the children to solve.
NRICH© call this the Low Threshold - High Ceiling approach where ‘learners develop into competent, confident mathematicians. The Low Threshold may mitigate against the development of maths anxiety by making sure that learners do not fail at the first hurdle.’ These attributes enhance the children’s ability to become more independent learners without the ‘stresses’ of struggling through problems.
We / the children identified the ‘first hurdle’ and addressed it before moving on. This removed the ‘stresses’ by allowing the children to gain a conceptual understanding each time, make connections in their learning, be successful and therefore feel more confident and positive towards becoming independent learners. It is also ‘fun’ to teach, especially with regard to allowing the children to explore with no pressure and not just ‘force’ them through the problem, struggling with each step.
Supporting children to make links
A commonly used method that I believe often results in children ‘struggling’ is the use of ‘RUCSAC’. This acronym, that represents the process of going through ‘steps’ to solve word problems, stands for: ‘Read, Understand, Choose, Solve, Answer and Check’.
The main issues for me are the first two words. ‘Read’ and ‘Understand’ are purely in the abstract and assume that children can both make the links to real life and have had previous experience of particular situations. As we saw in the ‘Grouping Goodies’ problem, this needs to be analysed to identify the concepts and associated to the ‘Concrete, Pictorial, and Abstract’ (CPA) approach as advocated by Jerome Bruner (1960):
As you can see, text is very sophisticated (abstract) and needs to be accompanied by the ‘iconic’, that is, items (concrete materials such as counters or cubes etc.) and pictures / drawings that represent values or situations. These need to be manipulated (‘enactive’) to explore and understand the concepts and linked to the ‘symbolic’ (numerals that represent amounts).
Please see my blog, When is number not a number? to explore the difference between ‘numerals’ and ‘numbers’.
Modelling and developing language
Derek Haylock and Fiona Thangata (2007) take this a step further and link it to the children making connections in their learning, including the use of specific language:
“Making connections in mathematics refers to the process in learning whereby the pupil constructs understanding of mathematical ideas through a growing awareness of relationships between concrete experiences, language, pictures, and mathematical symbols. Understanding and mastery of mathematical material develops through the learners’ organisation of these relationships into networks of connections.”
I believe that the word ‘communication’ should also be included across the network of connections where the children can communicate their ideas in a variety of ways. This background theory is all linked together in the approach that was used in the ‘Grouping Goodies’ activity that I described earlier.
Of course, maths is not just about solving word problems and I would like to share another example where I worked with a teacher and teaching assistants in Year Five. This was a whole class scenario where the children all worked together from the same ‘low’ starting point with the aim to reach the ‘High Ceiling’.
The teacher wanted to revisit ‘Dividing numbers up to 4 digits by a one-digit number using the formal written method of short division’ as she had identified gaps in their learning. Some of the children were making mistakes in their calculations. I suggested that we should revisit ‘remainders’ and the direct link to regrouping.
This was carried out and it quickly became clear that most of the children could describe the process of finding a remainder but several children had clearly not understood why each step of the process took place and what it meant.
The process of developing their understanding began with a simple revisit of remainders where we asked the children to explain the concept. This was done on large sheets of paper with multilink cubes as manipulatives which then moved to the pictorial and the symbolic. The children were allowed to explore and articulate their ideas.
The other key concept we wanted to address was where children were identifying a ‘remainder’ without realising that it could be included in the equal groups. The remainder had the same value as the divisor (see diagram below).
From here, most of the children moved to reasoning activities in division appropriate to Year Five such as:
The teaching assistant and I worked with the small group in the way that I have put forward and by the end of the lesson, they had a much clearer understanding of the concepts and were ready to move towards using formal methods and reasoning within them.
There are three reasons why I have shared this example.
- Firstly, the rationale of the Low Entry – High Ceiling approach can be used as a whole class, especially with regard to formative assessment without pre-conceived ideas of what the children can actually do.
- Secondly, the teaching assistant effectively addressed specific issues which also increased the children’s awareness and enhanced their independence. They all asserted that they felt more confident to move forward.
- Lastly, in my experience it is often the case that a teaching assistant might be asked to just practise the procedure of short division with the children. Obviously, this is important but to secure understanding of the concepts involved will improve their understanding of maths and allow them to make connections and subsequently increase their independence.
A unique position
My interest in working with teaching assistants stems from the late 1990s when I was involved in the making of a video with the (then named) Department for Education and Employment (DfEE) entitled ‘Working Effectively with Teaching Assistants’. Sadly, this is no longer available as it was recorded on video tape and was never digitalised. The rationale now, however, remains the same as it was then.
Teaching assistants are in unique positions to make a difference and need to make the opportunities worthwhile. Working in the way that I have analysed here makes the role thoroughly interesting and enjoyable with clear evidence of impact.
In writing about ‘Changing Mindsets’, Mike Gershon (2014) asserts that the development of independent learners, including developing positive attitudes towards mathematics and learning mathematics, can be achieved in a variety of ways, including:
- The use of a wide range of tasks and resources
- Enthusiastic teachers, with a 'can do' positive attitude
- Plenty of opportunities for children to experience success
- Hands-on approaches to learning
- The use of real life examples and exploring links with other subjects
- Offering positive role models of mathematicians
- 'Making it enjoyable' and celebrating achievements
He also considers the question: How do we develop confident learners who are able to work independently and willing to take risks? This is answered with the following suggestions:
Acknowledge all contributions positively and encourage learning from mistakes. Welcome wrong answers as the springboard to new understanding. Use positive language. Encourage independent and small group research and value different approaches and ideas.
Further professional development opportunities:
Digital training – reactivating lost learning and developing independence in mathematics with a focus on division.
These short courses will support teaching assistants to explore teaching techniques that feed the fluency and confidence of pupils towards increasingly independent learning. The video explores a number of focuses and includes pause-point activities. The accompanying learning log will help this.
The mathematical focus for this training will be the teaching and learning for division.
This training will be accessible until 21st July 2022
Booking links for:
Bakker, A., Smit, J. & Wegerif, R. Scaffolding and dialogic teaching in mathematics education: introduction and review. ZDM Mathematics Education 47, 1047–1065 (2015). https://rdcu.be/b95q9
Derek Haylock and Fiona Thangata (2007) Key Concepts in Teaching Primary Mathematics, Sage Publications Ltd.
Jerome Bruner (1960) The Process of Education, Cambridge MA: Harvard University press
Mike Gershon (2017) How to develop independent learners: http://beechwoodteachingschool.co.uk/wp-content/uploads/2018/11/Tes_strategies_to_develop_independent_learners.pdf
NRICH (2013) Low Threshold High Ceiling - an Introduction: https://nrich.maths.org/10345
Rachel Rayner’s blog: The ‘CPA’ approach