Arithmetic is the study of using numbers and working things out with them. Over the recent decades the meaning of ‘arithmetic’ has changed from being limited to achievement in standard procedures without full understanding to the one that establishes logical structures behind the numbers and their operations. Individuals living in the current century are required to take the initiative in making connections between mathematical representations and real-life problems. They also need to find appropriate ways of solving problems by working flexibly. These expectations have warranted teaching children arithmetic effectively through making connections, investigating number patterns, and identifying numerical relationships (Anghileri, 2006)
The aims of the current national curriculum in mathematics require pupils to become fluent (through frequent and varied practice) and develop conceptual understanding, reason mathematically along with being able to solve problems in a variety of situations. It also recognizes mathematics as an interconnected subject whereby pupils should be able to move fluently amongst different representations of mathematical ideas. The expectations are that the majority of pupils should be moving through the curriculum stages at the same pace. Children who have grasped a concept should be provided with enriching challenges through sophisticated problems (DfE, 2013). These principles have been reflective of teaching for ‘mastery’ approaches in high-scoring education systems of East and south-east Asia (NCETM 2014). Their high performance in the program for international student assessments (PISA) in 2009 and 2012, provided the rationale for introducing teaching for mastery in mathematics and other initiatives to deliver the national curriculum in England (DfE, 2019)
There is no precise definition of ‘mastery’, mention skemp- relational understanding and mike askew making connections there are various initiatives, groups, and organizations who have taken a lead in promoting mastery. Among these, are two high-profile Singapore-inspired initiatives called: Mathematics Mastery and Maths No Problem. These are based on the textbooks in Singapore and also provide professional development activities for educators (Boylan et. al. 2018).
School X (my first placement) was a two-form mixed community school with approximately 440 pupils on the roll between the ages of 3-11. The school has a very low percentage of pupil-premium children as compared to the national average. The majority of the pupils meet the expected standard in reading, writing, and mathematics however, the percentage of pupils achieving higher standards in Reading, Writing, and Mathematics is well below the national average. The school in recent years has not made sufficient progress in Mathematics and therefore under the initiative of the Maths leader, it has recently invested and rolled out the Singapore-based Maths No Problem scheme across the year groups.
Mathematics education in Singapore places importance on high expectations of all learners, covering the curriculum in depth allowing sufficient time, and giving every child the opportunity to develop conceptual understanding and skills. This is usually achieved through the use of mathematical models, high-quality mathematical talk, and thinking (Drury, 2014). However, School X still has a lot of inconsistency in teaching mathematics across the year groups; from some classes having mixed-ability seating to only some use of concrete resources or other forms of manipulatives.
For the purpose of this essay, I am going to look at using manipulatives, representations, and a Concrete-Pictorial-Abstract approach along with mixed-ability seating along with planning carefully crafted lessons as means to achieve effective teaching of arithmetic within the classrooms.
Using manipulatives, representations, and the Concrete-Pictorial-Abstract approach
The EEF report (EEF, 2017) identifies manipulative as “a physical object that pupils or teachers can touch and move which is used to support teaching and learning in mathematics.” They provide hands-on experiences in representing abstract mathematical ideas explicitly and concretely (Moyer, 2001) and mental images that pupils can use to contextualize mathematical ideas (Askew and William, 1995). Their usage in promoting mathematical understanding in order to make learning permanent is derived from works of researchers such as Zoltan Dienes (Dienes, 1969) who suggested the use of multiple representations of a concept as means of supporting pupil understanding, and Piaget (Piaget, 1952), who recommended that due to the lack of mental maturity in childhood to grasp abstract symbolic mathematical concepts, children need many hand-on experiences with concrete resources and pictorials for learning to take place. Skemp’s (Skemp, 1987) theory also favors the belief that pupils’ early interactions and experiences with concrete objects laid the foundations of abstract-level learning at later stages.
For instance, in the reception class of my Special Interest Placement school, we used multilink cubes to make staircases for numbers. We then used a puppet to jump one step up and one step down. The use of concrete resources in this scenario helped the children understand the concept of one more and one less from any given number and identify the pattern with regard to cardinal numbers.
Nonetheless, using manipulatives doesn’t often lead to an understanding of a mathematical concept. Sometimes pupils could use them in a rote manner with insufficient comprehension of the concepts underpinning the procedures (Hiebert and Wearne, 1992). Other times there is an inability in establishing the link of actions while using manipulatives to the abstract symbols (Thompson and Thompson, 1990). Manipulatives are merely tools and therefore their use within a lesson need to be purposeful, meaningful, and appropriate to create an impact (Carbonneau, 2013 & Askew, 2016). There are several important pedagogic principles for the effective use of manipulatives within the lessons. These are:
Both manipulatives and the activities for their usage should be matched carefully to the mathematical focus.
Pupils should be allowed to get familiarised with the manipulatives through play
Children should still be encouraged to make their own recordings on a whiteboard or paper, visualize and link the manipulatives to the abstract symbols.
The teaching around using manipulatives should involve discussions, comparisons, and appropriate usage of vocabulary to engage children with reasoning (Griffiths et. al., 2017)
Pupils of all ages can be supported through manipulatives. They should be used as a ‘scaffold’ to learning which could be removed to avoid pupil reliance on them (EEF, 2017).
By contrast, the usage of manipulatives was restricted to KS1 in school X. Nonetheless, I used colored counters to introduce a formal written (column) method of calculation for addition. I used these as each color could represent the individual column and the children could physically count and move them around to exchange when carrying out regrouping. In the first two lessons, having exposed the pupils to place value counters and charts for the first time, I modeled what I would like the pupils to do by using the counters at the beginning of the lesson. I then involved them in the activity through discussions around what I had done, and why regrouped using key vocabulary related to ones, tens, etc. In the next part of each lesson, a link was then made between what was done with the counters to the column method. This unpicking of the mathematical structure of addition with regrouping using the counters helped them develop a procedural understanding of the column method and they were able to justify how and why regrouping was involved in developing the conceptual understanding of the operation.
Representations
There is considerable evidence that concrete objects don’t become tools for thinking with ease (Askew, 2016). This is mainly because the mental images of numbers are linked with ordinal aspects whereby numbers are placed on a number line in order (Dehaene, 1999). The EEF report identifies number lines as “effective representations for teaching across both Key stages 2 and 3” (EEF, 2017). Although there is no specific guidance regarding the use of representations, as in the case of manipulatives, there is strong evidence that comparing and discussing a variety of representations can help pupils develop conceptual understanding and impact positively attainment (Ainsworth, 2006). The School X calculation policy has considerable emphasis placed on using number lines across the year groups however, was unable to see it being used within the lessons that were observed (School X calculation policy). Bar models, part-whole models, and arrays are some other forms of pictorial representation used widely across schools. The former two particularly in Singapore education and the Maths No Problem scheme. The versatility and flexibility around using a bar model in different areas of arithmetic and problem-solving across different year groups make it a really effective representation (Maths No Problem- Bar Modelling). My use of them to show the division
Concrete-Pictorial-Abstract approach
The bar model also draws on the Concrete-Pictorial-Abstract approach which has come to be recognized as an essential mastery concept in teaching arithmetic.
The approach stems from the work of Bruner on the development of children (Bruner, 1966). In his research, he proposed the importance of using concrete based on actions (enactive), pictorial with imagery (iconic), and abstract which is language-based (symbolic) representations for learning. Based on these, Singaporean education promoted the CPA approach whereby a concept is introduced with the help of concrete resources, it is then presented with the help of models or images, and lastly with the abstract notations of symbols (Maths No Problem- Bar Modelling and CPA approach). This approach to learning mathematics is at the heart of the Maths No Problem textbook scheme. The movement across the stages provides an overarching structure to the lessons and detailed attention is given to developing abstract understanding gradually through linking the stages (Hoong et. al. 2015). The aim for teachers is to help pupils become fluent in the symbolic mode which is at the heart of mathematics and students who merely work in the concrete and representational mode over the long term can hardly be doing mathematics (Ma, 1999).
Nonetheless, the fluidity expressed with regard to moving across the stages is something the Maths No Problem scheme of lessons does not make explicit. According to Hoong’s interpretation of Bruner’s theory (Hoong et. al. 2015), students who understand the abstract concept shouldn’t be always required to start their learning from the concrete stages. For instance, I have used concrete objects and pictorial representations as means of deducing reasoning of the abstract understanding among pupils rather than starting the lesson with the stages of the CPA approach. This however was not the practice among the other classes in the school who followed the Maths No Problem textbook lesson structure.
To summarise, incorporating the CPA approach flexibly meeting the needs of the learners in the classroom, and establishing connections between different manipulatives, representations and abstract ideas (Askew, 2010) would form an effective mechanism of teaching arithmetic for understanding.
Mixed-ability grouping
“Teaching in small peer-based groups has been used widely to enhance the active participation of pupils in educational settings for many years” (Ambreen, 2017). The relationship between group work and learning evolves from the theories proposed by Piaget and Vygotsky (Fawcett and Garton, 2005). The importance of peer interaction in the cognitive development of a child was a striking feature of Piaget’s learning theory. He believed that in group work, children can explain and verbalize an activity to each other and therefore get a greater insight into the problem through shared learning experiences (Gray and MacBlain 2015). According to Vygotsky (Vygotsky, 1978), the social and individual processes are interdependent in the construction of knowledge also called the sociocultural approach to learning. This approach of interactive learning enhances pupils’ cognitive development when takes place between more and less competent. This process of interactional change that occurs in cognitive development is termed as “the Zone of Proximal development” in the sociocultural approach (Cole, 1985).
These principles are applied through various forms of collaborative and cooperative learning opportunities in the educational system (Jarvis, 2005). However, until recently, these groups for learning were primarily based on the ability of the pupils. This was mainly due to the notion, that teachers could then offer work to children which was more appropriate for their level (Boaler, 2015). The 2008 review of research on grouping strategies in Primary schools drew unequivocal conclusions that showed the absence of academic benefits and rather negative consequences on child development (Blatchford et al. 2008). This ability grouping indirectly sends out a message that ‘ability’ is permanent and therefore develops a fixed mindset that relies on the belief that either pupils are smart or not (Boaler, 2015). Many other types of research have pointed to similar detrimental effects of ability grouping to both the lower and higher groups whereby the higher groups become fearful to make mistakes. This could also prevent them from attempting more challenging work (Boaler, 1997; Boaler, 2013a; Boaler and William, 2001; Boaler, William, and Brown, 2001).
This practice was still prevalent in School X, where children were either grouped within their class into tables of higher ability and lower ability, or pupils were grouped into top and bottom sets dividing them between two classes of the same year group. The Ofsted report for School X has identified the next steps as promoting pupil progress in mathematics, especially achieving higher standards for the more able pupils (School X Ofsted report). Based on the evidence from the previously mentioned research, it is probable to concur that ability grouping could be playing some part in hindering pupils making progress in mathematics in School X.
Researchers have consistently found that providing ‘Opportunity to learn’ plays a crucial part in school success and student achievement. When all students are provided opportunities to be exposed to advanced maths within classes regularly, the achievement outcomes improved for both the low and high-achieving students (Burris, Heubert, and Levin, 2006). This in turn promotes the idea of a ‘growth mindset’ whereby pupils believe that the harder they work, the smarter they get (Dweck, 1999, 2006)- paper on Singapore approach
With the strong drive in the new national curriculum promoting learning for all, there is now less emphasis on ‘ability’ grouping and greater on teaching the whole class. Whilst teaching for mastery it is vital to accept that with individual differences there will be variety in learning outcomes. These differences in outcomes will be with regard to the depth of learning achieved by pupils rather than the quantity they have learned (Askew, 2016).
One way of effective teaching in a mixed ability class at a whole class level requires providing the children with multi-level open-ended tasks also called ‘Low floor, high ceiling tasks. (For example, while teaching in School X, I would often give a number to children in their table groups, and each group had to show me different ways that they could make that number using the four operations of arithmetic). The teacher’s role within these lessons would be to facilitate high-end discussions by guiding the pupils to extend their thinking through the appropriate usage of key vocabulary. In addition, students should be offered choices of various tasks that address different levels of mathematics and the flexibility that doesn’t require them to always work on the same tasks (Boaler, 2016). In light of this, I introduced the chili challenge within my mathematics lessons. For every question, students were given choices of different tasks based on the heat of the chili. They could choose which tasks they would like to attempt. An example of this would be for the anchor task word problem, the pupils were given options of mild chili (which would be representing the problem in at least one way and then solving it), medium hot chili (which would be showing different ways of representing the problem, and different ways of solving it if any) and red-hot chili (which would be solving the problem and then using manipulatives to explain how they solved it). This provided opportunities for all children regardless of their abilities to be involved with the learning.
‘Complex Instruction’ is another effective strategy recommended for teaching a mixed-ability class (Boaler, 2016 Cohen and Lotan, 2014). This strategy involves assigning roles to each member of the group and encourages pupils the importance of group talks through good questioning, valuing different ideas and approaches, connecting a variety of methods, and calculating and evaluating solutions (Boaler, 2008).
The Singaporean textbook approach, which was used at School X, recommends the whole class moving at broadly the same pace within the lessons. The beginning of the lesson is usually a contextualized open-ended problem that requires whole class engagement and participation. The collaborative work within the groups involves the use of manipulatives, representations, and conversations around possible solutions. This helps develop a classroom whereby there are opportunities for discussions amongst children of all abilities enhancing their understanding of the mathematics on hand. The scheme has some positive impact on developing growth mindsets for effective whole-class teaching. It further gives the basis for teachers to spend more time exploring Maths as every lesson follows the same structure of exploration, reflection, and practice (Boyd, 2017). This scheme approach does not place emphasis on task or content differentiation thereby avoiding in-class grouping through prior attainment (DfE, 2014).
Along with the ones discussed in depth within this essay, there are other aspects that contribute to the effective teaching of arithmetic. Some of those are planning lessons in a sequential manner (planning for progression) in order to cover the smaller steps needed to introduce the big ideas, within each such lesson using carefully selected few examples that extend pupils thinking rather than lots of different ones (variation theory) and the teacher talk within the lesson which facilitates learning through questioning and verbal reasoning. In addition to the teaching practices, creating a classroom environment that promotes challenges, and embraces struggle and mistakes is equally important to ensure good mathematics learning for all.
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