# The E.L.P.S Theory and Connectionist Approach – Comparing Two Theories on How Children Can Best Learn Mathematics

The E.L.P.S Theory and Connectionist Approach – Comparing Two Theories on How Children Can Best Learn Mathematics

Overview
There are numerous theories about how children can best learn mathematics. I will compare two of these: the E.L.P.S. theory (1984), and the connectionist approach.

Introduction
Mathematics is widely regarded as an abstract subject (Liebeck, 1984, p.14). To help explain the sequence of abstraction that children need to forgo to truly understand a mathematical concept, Pamela Liebeck (1984, p.16) devised the E.L.P.S theory:

E – Experience with physical objects,
L – spoken Language that describes the experience,
P – pictures that represent the experience,
S – written symbols that generalise the experience.

The connectionist approach, alternately, places its emphasis on pupils making connections from one context of mathematics and, drawing upon that knowledge, applying these to the particular area of maths that they are learning.

Similarities
Experience
Both theories share similarities. Most obviously, both encourage pupils to draw upon their previous experiences when attempting to understand new concepts.

Atkinson (1992) stated 'Maths is regarded as a powerful tool for interpreting the world and therefore should ideally be rooted in real experience across the curriculum' (p.13). This sentiment has since been echoed by Anghileri’s (1995), who urged:

‘. . . what we need to do is to start with the real problems, and work from them to abstract representations, not the other way around' (p. 28).

As presenting children with data-orientated (abstract) worksheets would not give pupils any connection to the ‘real world’, I instead had children measure objects around the classroom (fellow pupils, length of feet, playground area, etc).

Through this, my pupils physically experienced the various properties of measurement – through play, touch, feel, and comparison (Liebeck, 1984, p.16) – in a way that related to the real world. This is a fundamental part of Libeck’s message.

A connectionist teacher will use past experiences by drawing connections to those of his pupils, thenceforth applying these to assist them in learning new skills.

This is approach works on the logic that, by highlighting connections that a pupil can then relate to a new context, pupils can easier achieve an understanding of that context (p.60). Lucas (2001) echoes this sentiment, even going as far as to say:

'In order to support mathematical development we should find ways to enable children to make connections between what they already know and what they are learning' (as cited from Pound 2006, p. 32).

An example of connectionists using experiences in the classroom is when a teacher urges pupils to calculate the change due in a monetary transaction: the teacher draws upon the pupils’ knowledge of subtraction, and applies it accordingly.

Language
Once a pupil has drawn upon their relevant experience, Liebeck (1984) states the next important stage in the sequence of learning through abstraction is the use of language.

If a child has an opportunity to describe, through verbal language, the experience that they have had, it broadens their understanding and presents an opening to discuss and consider any problems that they may have encountered (Liebeck, p.16, 247).

Similarly, connectionist teachers also place an emphasis on verbal interaction, believing, as this allows pupils to leave primary school with confidence and competence in explaining their views and challenging the theories of others (Askew et al, 1997, p.11-14).

Language is seen as such a key element across the board that Atkinson (1992) describes it as the most effective tool when seeking to develop children's mathematical concepts (p.13).

Pictures
Once pupils have been encouraged to verbally describe their methods, Liebeck (1984) introduces the next stage of learning as incorporating pictures or diagrams (p.16) – as these 'clarify the essentials of a problem at its outset' (p.247).

Recently, I was teaching doubling to a less-able pupil in my year two placement class. He was struggling with this concept, so I tested Liebeck’s theory by devising an activity that required him to count a group of objects in picture form.

Once he’d calculated the number of objects in the group, I asked him to copy, exactly, the group of objects on the other side of the picture. Then, he could to count up both groups, and thus understand that a group of, say four, when doubled, equals eight.

For this particular pupil, pictures greatly assisted his understanding of the concept.

[Is there anything about connectionists and symbols??? If not, move this bit to the differences section]

Symbols
The Final attribute pupils need to apply to fully understand a concept is the use of symbols (Liebeck,1984, p.16). For pupils to solve and apply their knowledge in a simplified written form, they need to know and recognise the appropriate symbols (Liebeck,1984, p.247).

Liebeck (1984) is correct in stating that in order for pupils to understand mathematical concepts in abstract from, they must have understanding of the relative symbols. Yet teachers must judge when it is the appropriate time to introduce symbols to a pupil.

Research suggests that teaching conventional symbols to children at a very early age can be ineffectual (Hughes, 1986, as cited by Atkinson, 1992). This suggests that if children are not yet confident in using the conventional mathematical symbols, then devising their own symbols to represent a particular mathematical notion often gives them a greater understanding of a concept. Furthermore Atkinson (1992) believes that Hughes research may indicate that there may be a degree of insignificance when teaching very young children conventional symbols.

Qualities of the Connectionist Theory
The value of enabling children to make connections across mathematics is that it gives them a broader understanding of mathematics as whole.

For instance, when teaching multiplication, we can explain that it is essentially repeated addition (G. Bottle, 2005, p. 61). By explaining it like this, pupils can instantly relate to a skill previously learnt, and apply it to the new area of learning.

Briggs and Davies (2008,) sum up this aspect, asserting, ‘Children should be encouraged to find relationships between activities undertaken and make connections about the different aspects of mathematics' (p.89).

While Liebeck’s ‘transmission’ theory centres around the teacher giving precise methods for pupils to learn and use, connectionism requires pupils to formulate a strategy for calculating, with the teacher responsible for assisting pupils in refining their techniques (Askew 1997).

However, Askew he doesn't completely disregard other, competing approaches. Instead, he argues that the connectionist approach absorbs represents their best attributes, while not possessing their weaknesses (Askew et al, 1997).

Simply put, pupils are not taught just one specific method of addition (transmission method), and are not left largely unaided (discovery). Connectionist teachers therefore discover how individual pupils learn, and react by teaching them accordingly.

The connectionist approach also encourages pupils to share their methods, as opposed to being restricted to specific techniques; which transmission teaching would entail.

Connectionist teachers are not seen by their pupils as the only approved outlet of knowledge. Instead, the teachers value their pupils’ own individual methods and explanations.

In the connectionist way of teaching, if a pupil devises a mathematical method that is suited to themselves but not the teacher, the teacher could simply aid the pupil by advancing or (if necessary) modifying their techniques.

Finally, the connectionism encourages an inquisitive mind throughout the whole of a pupil’s life: learning through techniques they themselves have had an influence in creating, and are therefore comfortable with.

As a result there is a far greater possibility of pupils reaching the potential for gaining full understanding of a mathematical concept through connectivist abstraction.

Differences
Liebeck states that children will learn most effectively when following the E.L.P.S sequence in a hierarchal order (p.16). She claims that a textbook or worksheet will only engage children with the last two stages of the sequence, therefore neglecting experience and language where Liebeck asserts children need to begin when initially learning a new concept (p.16). Haylock and Cockburn (2003) reinforce Liebeck (1984) by agreeing that the stages of experience, language, pictures and symbols are detrimental to children’s learning (p.2-3). However, Haylock and Cockburn (2003) believe the order at which each stage should be introduced to children differs to Liebeck (1984). They use the following diagram (2003, p.4)

When referring to the above diagram, Haylock and Cockburn (2003) state that any of the four aspects referred to above ‘represents a possible connection between experiences that might form part of the understanding of a mathematical concept’ (p. 3-4). Haylock and Cockburn (2003) effectively apply the E.L.P.S theory and the connectionist theory together, and draw upon any appropriate connection, whether it be explicit language, an experience, a symbol, or a picture that will aid a child to understand a new concept. In a very similar sense to Liebeck (1984), Askew (1998) identifies the importance for children to be confident in the following modes of representation: - calculation of figures, words, pictures/diagrams, practical settings, and realistic content (p. 77-78). However Askew (1998) goes along Haylock and Cockburns line of thought by arguing that it is just as important to be able to move between the different modes of representation, as it is to be confident in their application (p.78). Hence (like Haylock and Cockburn 2003) Askew (1998) doesn't always believe there should be a specific routine in the teaching of mathematical concepts

Potential difficulties that present themselves when applying E.L.P.S or the Connectionist theory.
Despite connections in Mathematics being ever evident, children often need guidance from the teacher in identifying these connections, 'Making connections does not usually happen by accident and many children do not see the connections for themselves; they need to be made explicit. It is therefore incumbent upon the teacher to plan to highlight these connections.' (G. Bottle,2005, p.60-61).

Using materials with the intention of creating a experience that will assist pupils in the sequence of abstraction (Lieback, 1984), is an area where teachers are often perceived to be ineffective (Drews and Hanson, 2007, p.26-28).

In terms of using resources in experienced based learning tasks, Drews and Hanson (2007, p. 28) highlight the importance of teachers having a specific purpose for the resources to facilitate. Drews and Hanson go on to state 'In itself, the physical exploration of concrete materials will not lead to children 'discovering' mathematical concepts' (p.27). Drews and Hanson (2007) explain that mental activity that underlines the worth of the physical activity is the vital factor when developing mathematical understanding (p.27). An example of good practice when carrying out the abovementioned could be a teacher discussing with their pupil the relationships between the experience that they have had with the physical materials and the abstract ideas (Drews and Hanson, 2007, p.27). In supporting Drews and Hanson (2007), Muir (2007) states 'In order to help students develop mathematical understanding, the teacher must select examples that enable students to construct accurate knowledge about the concepts presented.' (p. 521).

Teachers need to be aware of the need to highlight the numerous connections between the different concepts of mathematics G. Bottle (2005, p 60-61). Despite connections in Mathematics being ever evident, pupils discovering connections between concepts independently is a rarity (G. Bottle, 2005, p.60-61). Consequently, these connections need to be presented to pupils precisely by the teacher (G. Bottle, 2005, p.60-61).

My experiences of applying the E.L.P.S theory through a series of mathematical lessons.
The class that I was working with were a year two class within a private school. The topic in which I was teaching and attempting to apply the E.L.P.S theory to was solving numerical problems involving money. This is covered in the national curriculum, key stage 1, Ma2 (choose sensible calculation methods to solve whole-number problems ((including problems involving money….)), drawing on their understanding of the operations). Due to the class following the Abacus format in mathematics, the lessons pursued their sequence of topics, and therefore were based on solving problems concerning the addition of money. However, two of the higher achievers progressed onto calculating change.

I believed it to be important to discuss with the class teacher the previous experience the class had had within the classroom related to money. My findings were that in year one they had been introduced to calculating small amounts of money, mostly the adding up of two coins. The class teacher believed that very few of the class would have had experience with calculations dealing with amounts above one pound. As a result of this conversation, when applying E.L.P.S to my teaching, it was key to build on the past experiences pupils had encountered, recap and develop language associated with money, provide pictures that will help children to recognise and learn the process's of calculating monetary values above £1, and finally teach children the correct principles of monetary notation (symbols). Further more, since the calculation of money shares numerous similarities to addition, I will ensure pupils can relate to, and see the connections between the two concepts.

Interview
The class that I was working with were a year two class within a private school. The topic in which I was teaching and attempting to apply the E.L.P.S theory throughout in, was money (predominantly the usage of coins).

Before beginning the topic I interviewed two pupils who were differing in attainment levels, and attempted to gain knowledge of their current understanding of coinage. I examined their awareness of the variety of coins in use, and consequently the amount of experience they have had with money. I also tested their ability to look at a sum and verbally declare the amount, and finally, their understanding of how to record various amounts in writing using the appropriate notation.

The higher attaining pupils (named Sam) had had more prior experience in using money than the lower attaining pupil (named Cece), and it would seem that through this prior experience Sam had a greater pool of knowledge than Cece (see appendices 2). Sam was familiar with the names and values of all coins, whereas Cece had no knowledge of the £2 coin, and only remembered some of the others when prompted by a picture card. The main experiences Sam had with money were through having a designated day where he had the opportunity to purchase sweets. He explained (appendices 2, p.2) how he always had an amount totalling 30p, however there were always a different variety of coins that equalled the amount. Whether or not this was his mother purposefully trying to provide opportunities for her son to develop his knowledge with money, I am not sure. What is for certain is that these experiences had worked towards enhancing Sam’s awareness and knowledge, as well as developing his confidence in using money. Cece on the other hand, who had little experience to speak of, had obvious holes in her monetary knowledge. The aforementioned would correspond with Liebeck (1984) by suggesting that relative experience is fundamental in the initial learning stages of a new concept.

Both Sam and Cece showed some confusion over writing the correct monetary notation. Sam knew how to write the pound sign, but when asked to write one pound thirty down, he wrote 1.30£. Cece could recall the correct symbol for pennies, but not pounds. Thus, these findings concur with Liebeck (1986, p.16) that learning the correct symbols is often the final stage before being able to fully understanding a concept.

Lesson 1 (appendices 3)
The learning objectives for the first two lessons were as follows:

• Learn to recognise all coins and begin to use £.p notation for money.
• Find totals of sets of coins: relate to adding three + numbers.

My initial objective was to understand the differing range of knowledge that existed throughout the class. In order to do this effectively and efficiently, I found it hard to create a way in which to achieve this by commencing the topic with an experience based task, as the E.L.P.S theory would prescribe. Consequently, I started the topic by presenting to the class large pictures representing each individual different coin. I asked pupils to raise their hands if they could name the coin being displayed to them. Hence, the task dealt predominantly with the language and picture stages of E.L.P.S, as pupils had to use the correct language when naming the specific pictures, although it could be argued that pupils were drawing upon previous experiences and applying it to their answers. The results were that when shown pictures of coins pupils were nearly always correct in naming the right coins, and correct in the language that they used, i.e. '20 pence', or 'two pounds'. This told me that Pupils had memorised much of what they had been taught in year one.

Once this introduction was complete, I reverted back to the start of the E.L.P.S sequence, and gave the pupils an experienced based task. When working in pairs, pupils had to use the plastic money (using physical resource) that was available to them, and place their coins in the correct order, starting with the lowest value coin, and working up to the highest valued coin. At this stage, when explaining the task that I was setting, I encountered an error with the language that I used to the children. I explained this activity using the word 'value' in the explanation. A pupil subsequently asked me 'what does value mean?', I had presumed that pupils would have an understanding of the word. However, when asking the class if anybody could give me an answer to what it meant, no child volunteered. I can reflect that the experience set out for the children naturally lead pupils on to learning about relevant language in relation to the context. This activity therefore matched the sequence of Liebecks (1984) E.L.P.S theory.

In the next activity I was questioning the class at random on how many pennies there are in certain coins. In this activity I aimed to establish pupils understanding in the concept of applying monetary notation to numbers. Pupils found the activity straight forward until the £1 coin and the £2 coins were presented. Not all pupils had identified that there are one hundred pennies in a pound. As a result of this occurrence, I improvised and began a short activity where I asked pupils to make one pound using any coins they liked (apart from the £1 coin). In this I was attempting to provide an experience that would help children to primarily learn the value of a pound, but also allow pupils to realise that there are numerous ways in using coins to get to an amount. In further support of this activity, and in a connectionist style of teaching, I had the class verbally recite the ten times table (prior observation informed me that the class had firm understanding in this area), due to the larger valued coins are all multiples of ten, they could apply this knowledge to the abovementioned activity.

The next activity was again devised as another physical experienced based task using the plastic money. On the magnetic whiteboard I drew a purse, inside the purse I stuck in a number of coins, and the children had to calculate how much money was in the purse. I told children I wanted them to acquire the coins shown on the whiteboard from their plastic money, and use the coins to help them with the calculation. At first I only placed two coins in the purse, then I progressed onto three and four coins. The activity was an activity that is relative to children's' life. Adding up coins whether it be in their piggy bank or purse is an experience that they would either have had already, or will come across in the near future.

On the whole I thought that pupils gained some valuable experiences throughout the lesson that aided them in their learning. The use of physical resources within these experienced based task was also very effective. Despite the introduction not being specific to the E.L.P.S. sequence, I felt it was essential to gain an initial assessment of pupils. Their was evident progress to be seen throughout the lesson with the lower achievers in the class, I believe up until nearing the end of the lesson, the higher achievers were not being adequately challenged, however, I do feel that their prior knowledge was being consolidated.

An area where I felt my lesson was weak, was in my lack of emphasis on the importance of dealing with money in everyday life. In my failure to address this, I believe I may have been unsuccessful in engaging pupils to achieve high attention levels. (quote)

Another aspect of the lesson that was potentially problematic, was the size of the pictures that I used representing the different coins. Due to the size of these (laminated A4) children may have found it hard to draw the comparison to that of a standard sized coin. It may have been more insightful to use real coins for the demonstration, giving pupils opportunity to relate the experience to that of real life.

Lesson 2 (appendices 4)
From lesson one I had assessed that all pupils had good knowledge in terms of the recognition of coins, however, pupils still had to develop and improve in the use of £.p notation, and the adding of three or more coins.

With the learning objectives stating that the aim is to 'begin' to get pupils using the correct monetary notation, and the E.L.P.S. theory prescribing that learning symbols is the last stage of the learning sequence, I subsequently (in this lesson) didn't pay too much attention to the accuracy of pupils marking, as long as there was an obvious attempt.

I began the learning with discussing with the class why it is important to learn about money (ideally I would have done this at the beginning of the previous lesson). I explained that very soon in their lives (if not already) they will be required to use money to pay for things, such as food and drinks in the tuck shop, and then, when they are much older, they will have to pay bills etc. The class discussion engaged all the children who were keen to express why being able to deal with money was important. I believe as a result, levels of focus were greatly increased in comparison to the previous lesson.

The first activity was an interact problem solving game on the interactive board. The tasks within the game were predominantly word problems involving two or three coins. I would read the problems out to the class and require them to solve the problem individually. As always, pupils were encouraged to use their coins in their workings out. The problems provided used language which was directly related to the learning objectives, for example words such as spends, pound, buys, penny, how much, spend in all.

The class then went onto solving some picture orientated problems. I began the higher attaining pupils on the activities set out in pupils Abacus textbooks, these were set specific to the learning objectives and I was confident that they would complete them efficiently, hence I had more challenging tasks prepared. For the lower attaining pupils I wanted to keep the elements of addition realistic to their current attainment levels, I therefore started them on task that were relatively basic, and then progressed them onto the Abacus tasks. As previously mentioned, the activities set were rich in pictures