2 6 VALIDACION DE LA PROPUESTA

U076: It has made me discover that maths is fun when you understand what you are doing.

This view defined for me involvement in activity based on pupils’ perception of learning as ‘pleasurable’. This relates to the emotional dimension of pupils in classrooms; yet it is often overlooked that the pupils’ emotional state may affect the quality of mathematical learning. Using computers can allow pupils to explore ideas, exercise responsibility for their reasoning, hence facilitate enhanced understanding in the changed classroom environment and generate enjoyment of their learning. Figure 4.8 shows that 67 % of all the pupils (181) reported ‘enjoying’ the change in their learning due to the presence of computers in the mathematics lessons. The pupils clearly attributed their enjoyment to the use of the Grid Algebra software on computers.

K150: Grid Algebra is easier to understand than oral hearing. It makes you enjoy maths and you want to do more. It has made me quicker in thinking. It has made me love mathematics more and enjoy the lessons more.

R235: I love computers very much and it is easy to understand when I am using it to learn mathematics. I learn by doing something I love.

R225: It’s enjoyable and enables one to understand algebra easily. It has made learning algebra easier and more enjoyable.

Figure 4.8: Distribution of pupils’ views on the experiences in ICT-enhanced sessions

For other pupils, their ‘fun’ seemed to be due to their increased understanding of algebra.

U104: I never knew how to carry out algebraic expressions. Now I really enjoy calculating.

P184: It makes a topic that felt so difficult, understandable. It made me enjoy algebra while doing it.

K154: I got to understand a huge bit of algebra that was giving me a problem. This includes how to simplify, writing expressions and especially the use of signs as I was never able to tackle questions with positive and negative numbers. It has made me feel that mathematics is easier and even more fun as you learn more.

‘Understanding more’ of algebra was something the pupils were clearly aware of (see Section 4.4.4). Figure 4.8 indicates that many attributed their ‘enjoying’ to increased connectivity of algebraic concepts.

R220: Using computers is very fun and when something is enjoyable, you understand easily. It is a more enjoyable way of learning and one actually sees the process of doing the sum.

K129: Because for one it was fun and interesting thus making me understand better. Not only did we learn mathematics but also computers.

For some pupils, mastery of the software was a gradually transformative process.

U075: It was difficult at first but with more practice I was able to understand and work more easily, even on my own. It helped me learn faster with fun in mathematics.

P206: At first one may not understand well therefore just follows the arrows without knowing why you are adding, subtracting, dividing or multiplying. With more practice and knowledge of how (Grid Algebra) works you get to understand algebra much better.

These pupils appeared to suggest that using the software enabled them to move from one form of learning to another; from procedural, where they simply made the movements across the grid without understanding why, to relational, once they had mastered using the software. It seems that pupils’ mastery of Grid Algebra may have facilitated more effective pupil learning of algebra. This idea was emphasised by one teacher.

T3: Conceptualizing in algebra has actually been taken to some levels higher, like 2 or 3, or more, more than that!They know the terms: that a term in algebra is differentiated by a sign, either positive or negative; these pupils know that!

This teacher’s view was amplified in several pupils’ descriptions of their developing formal notation and use of mathematical terminology. I responded to some questions on ‘variables’, ‘coefficients’, ‘terms’ and ‘constants’ from textbooks the pupils had attempted. This seemed to help the pupils use the new vocabulary when they engaged in small-group discussions and writing.

K128: It has benefitted me since it has helped me understand from the smallest bit. It helped me understand how mathematical terms are written in algebra e.g. instead of

.

P182: I have a chance to at least understand mathematical signs (brackets, division line) and learn new terms like inverse, factorise.

U089: It has helped me learn a lot of what I did not really know before like writing expressions, use of brackets through Inverse Journeys.

K152: Doing algebra using a computer is much easy to understand... It helped me to understand how to multiply the signs in and out of brackets.

For some pupils, it was the emphasis on communicating mathematical ideas when writing.

P185: It helped me in learning how to arrange my calculations.

G002: I’ve known how to arrange my work neatly and present meaningful work.

G043: I can now write my mathematics work in order without jumping some steps, making my work tidy.

U089: It has helped me learn a lot of what I did not really know before like writing

expressions, use of brackets through Inverse Journeys. It has really helped me improve in my grade. It has changed the way I learn maths what I did not know of I know now like how to arrange my work when writing.

A sense of ‘learning while playing’ evoked in ICT-enhanced sessions was alluded to.

P165: For a person like me, I had a lot of problems with algebra but now I have just found it easy and exciting because it is all about playing with numbers.

All pupils were highly absorbed; some showed positive emotions and concentration on tasks. They worked collaboratively through difficulty levels. When the sessions ended, many pupils were reluctant to leave the room: pupils asked to be allowed to continue with the activity, as one said:

R233: I love it! (Dancing) It is quite addictive; hard to let go…

Whilst most pupils seemed very absorbed in solving the Grid Algebra-based tasks, one pupil inquired about acquiring the software for personal use.

P205: It would be great to have it at home instead of just having computer games! I can play but at the same time I’ll revise my work and build confidence in maths even as I relax.

This pupil described the ICT-enhanced mathematics sessions as “different from the usual boring class lessons”. Other pupils working in the adjacent group asked me whether a mobile phone ‘app’ for Grid Algebra was available. I honestly admitted to having no such

information. The pupils’ enjoyment of ICT-enhanced sessions was echoed in teachers’ observations such as:

T1: One thing we can say is that there is a lot of fun in these mathematics lessons compared to the classroom teaching.

What emerged as clear was that the pupils appeared to develop a sense of ‘love’ and ‘enjoyment’ in their learning during the ICT-enhanced mathematics lessons.

K141: Because it assist(s) one to understand algebra in a fun and exciting way.

U082: It has made me love mathematics and I have enjoyed a lot. It has enabled me have more love in mathematics than before.

P200: Because you have fun as you are learning at the same time. The way I understood Integers is different from the computer as in the side of arithmetic.

K156: It’s enjoyable and during that period one actually learns a lot. It’s really fun calculating maths there! It enabled me understand better.

U065: It makes algebra seem simpler than when it is taught in the classroom. It made algebra more fun and exciting.

Some powerful emotions and effect on the pupils’ concentration were evident during the timed Task 13:‘Make the Expression (letters)’. This particular task elicited reactions of increased chatter from pupils in all the five classes; observed norms of classroom protocol and social inhibition fell away whilst the pupils worked with great excitement in battling to ‘beat the clock’. So great was the noise generated in learning activity that several members of the senior management team rushed to the laboratory only to find pupils engrossed in their learning, totally oblivious of attracting undue attention. Every pupil remained very much focused on brainstorming in their small groups the possible movements to make across the grid. I saw pupils making physical actions as they spoke, and trying out different routes, to create prescribed expressions within time limits imposed by the software. I found it difficult to concentrate on or hear what some members in any one group were telling each other since my attention focused on ensuring the classes did not disrupt the rest of the school: these pupils were that noisy!

P180: It made me pay a lot more attention since once the questions passed, it brought more questions and somehow harder.

K148: They were very enjoyable since they were in game so you had to be on your toes so as to get any sum correct.

The pupils stated that albeit being ‘fun’, the software-generated tasks were in no way simple. With the variation in prior computing expertise attributed largely to the diversity in schooling background (see Table 3.2 in Section 3.2.3), some pupils seemed to value working and discussing in groups.

R251: We got involved in group discussions and help each other with the work given which most of the time does not happen in the classrooms

K160: We get to learn to do mathematics using a computer and together, which makes it easier to discuss and understand.

Great value was placed on the ‘community of learning’ facilitated by collaborative working since many pupils developed their confidence and skills of communicating mathematically as they articulated their spoken ideas.

G023: I also liked it because you learn how to use the software and get to listen to each other as you work together on sums and figure out expressions.

U077: Grid Algebra has helped me understand algebra because I had problems in understanding and reasoning.

In summary, many pupils clearly appreciated using the Grid Algebra software on computers in realising enhanced enjoyment of learning, conceptual understanding and connectivity. The task of distinguishing specific factors that engendered pupils’ fun was not easy. It is possible the pupils’ developing belief that computers make learning more ‘enjoyable’ may have been responsible for increased accessibility of ‘difficult’ algebra.