This theme revolves around verbalisation when using ICT-enhanced opportunities for active learner participation in mathematical activities. Many pupils indicated an awareness of the role of the computer in motivating them to take control of their own learning.
Figure 4.2: Distribution of pupils’ views on learner participation in the computer-based activity
This alluded to increased learner agency in the mathematics-with-ICT environment. These pupils stated feeling more engaged.
P190: I get to understand more when using Grid Algebra and I could also apply that even in my book. It is fun to use the computer to study. You get to study and became active. Some people do not get active in class but there (in the lab) one is totally active so that you may understand the steps one needs to take.
G053: I prefer to learn algebra through Grid Algebra. It takes me step by step to learn a certain expression. And it also involves my participation as a student.
0 20 40 60 80 100 120 Curiosity & Prediction
More Involved Teacher-Centred Pedagogy Activity 'Practical' Visualisation Valued Immediate Feedback
Learner Agency
No. of Pupils 113 22 38 19 31 22K146: We are used to teachers in class every day and writing. It is good to explore
technology. It is easier for me when I master the concepts used in the computer then I apply them on paper. It opens up the mind.
G022: It is also easy to operate the computer. I got to do algebra practically thus increasing the rate of my understanding the topic.
P177: It is important to move with technology. It has boost (sic) my thinking by making me find solutions for myself.
These views resonated with those expressed by the HOD-Mathematics (see Section 4.4.6.5); they illuminated pupils’ inactivity in teacher-directed instruction. The views echoed claims that I made about the Kenyan context where teacher-talk usually dominated in mathematics lessons (see Section 1.1.2). Other pupils apparently welcomed the increased opportunities for classroom dialogues around learning and involvement.
G044: We so much just sit in class, the teacher would come teach and write on the board. We are not free to talk to our desk mates and ask questions but at least after this, we changed our learning zone and were able to discuss.
P193: It opens up one’s mind so as to think and know how to operate different equations. It changed the way I learnt because in class, sometimes I cannot understand how the teacher came up with different solutions, but Grid Algebra explains and sometimes shows arrows on how to operate different equations.
R254: It was easier to work with a computer than by hand. Learning changed since I understood well than when a teacher explains it on the board.
These statements pointed to pupils’ valuing of the emphasis on articulation, variety and group discussions as learners negotiated their mathematical understandings. Some pupils expressed feeling excluded and demotivated in their regular mathematics lessons.
R223: Having these lessons has helped me a lot in my class work, encouraged a lot to put effort in my work since I was previously very frustrated. We got to explore learning using computers and it made mathematics fun.
K117: Before Grid Algebra was introduced, I hated the topic in mathematics because I understood nothing! But when it was introduced, I like the topic because I enjoy doing it and also I can understand what I am taught.
It is arguable that the passive role, to which these pupils were usually relegated, engendered feelings of frustration, lack of understanding or even dislike for learning mathematics. Figure 4.2 illustrates that 42% of pupils (113) valued trialling and re-trialling to satisfy their
curiosity and to test their predictions as they explored and experimented on the computers.
R264: The way we learned has changed in that Grid Algebra has made us understand more about algebra since you experiment by yourself making you get the concept well.
U101: This has helped us know how algebra is worked out. You are able to see and get to know how algebraic expressions are formed.
P181: It has helped solve my many algebraic puzzles. At least now I know how to manipulate different expressions. Algebra used to be a problem for me. I’ve been able to now understand how and why the final answer comes to be.
U060: It was beneficial since it helped me know how to calculate using computers. It helped me to be able to believe in the answers I give during calculations.
This suggested that kinaesthetic activity was contributing to the pupils’ cognitive
development. Placing the ICT tool in the pupils’ hands appears to have facilitated individual construction of mathematical connections; and hence ‘deep learning’ and ‘relational
understanding’.
P211: I got to understand algebra in a different way, as journeys on a grid and solving equations as inverse journeys, which was much easier and more fun.
R256: Grid Algebra helped me understand deeper and deeper since algebra was a problem from primary but now it isn’t a problem again. It helped me to understand algebra and the steps of solving any question from algebra.
Others described the learning activities as ‘practical’; they learnt more by ‘doing’.
P183: It helps us understand more fast (sic) since we do it practically. Before we learnt in class where most people didn’t understand but Grid Algebra is easy to understand.
G049: Because the Grid Algebra helps you to work practically and helps you to be active unlike in just oral which we do not understand or is even boring. It has instructions and examples which made me understand and brought reality to the ways of working.
P165: We got to understand algebra better since it was practical than when taught in class.
U095: It has enabled me to see the real sense of mathematics in calculations.
Many pupils valued understanding ‘why’ in mathematics over getting ‘correct’ answers.
P166: Grid Algebra gives us chance to understand the concepts of doing algebra before tackling the actual question. Before, I did not know how I came up with the answers I got in algebra but now because of the program I am able to understand fully.
K117: I like the topic because I enjoy it and also I can understand what I am taught. Before Grid Algebra was introduced, I passed but I didn’t know what I was doing since I practised the spoon-feeding method from teachers.
These pupils’ responses depicted some form of ‘anxiety’ with the focus on procedures rather than understanding in their usual textbook-based and teacher-directed instructional practice.
T3: My happiness is when I bring a concept, in my own way, I give them the steps, and then I see the (pupils) working step-by-step, and they produce, that is my achievement, what I love!
It seems that Teacher T3 valued transmission of knowledge through exposition. Grid Algebra
use emphasised the structural relationships, hence consolidated the symbol convention.
T4: It has been so difficult for them (pupils) to know ‘how do I introduce this bracket or why or where?’ But when using this (pointing to the Grid Algebra program on the screen), they see the way the bracket is being introduced for them, they understand why it is so, so that even when it comes to written work, it becomes very easy, and the (pupils) can do it just like they saw on the computer!
This indicated teacher awareness of pupils’ problematic access to formal notation. Renewed focus on the symbolic language provided a more meaningful introduction in algebra.
T3: The use of brackets was easier; it taught the pupils that when you see brackets, whatever is inside, when you open, you multiply with what is outside. We are not having an issue, no hitch on that.
Other pupils underscored the scaffolding offered to their conceptual understanding by various features of the ICT tool as they learned.
G043: It shows us the step by step of how to get to an answer rather than having some examples with teachers and you wouldn’t know why.
P179; It’s a new way and easier to understand compared to when you are only taught on paper. Grid Algebra provides answers, clues e.g. routes which help one understand better compared to when you are doing it without any clues.
I encouraged all pupils to pause and think about the questions and feedback whilst discussing with each other. In each of the five classes, increased requests for assistance indicated pupils’ confusion particularly with vertical movements and deciding what to multiply with or divide by in Task 7: ‘Find the journey (letters)’ during ICT-enhanced session 2. Apparently, they had forgotten how the software works. I attributed the difficulty to a week-long hiatus of sessions. I was prompted to recapitulate the association between grid movements and mathematical operations, directing the pupils’ attention to the fact that the rows were pre- determined. Whole-class discussions appeared to refresh the pupils’ mastery of the software. Most pupils worked successfully through levels 1, 2 and 3: some managed level 4 with increasing confidence. The pupils considered their peers’ actions and listened to
contributions; they squealed with delight at the interactivity. Towards the end of each session, I drew the pupils’ attention to Task 25: ‘What is the expression?’ which they received very well; the activity gave pupils something to anticipate in one week’s time.
R222: It is kind of practical and easy to understand for example inverse journeys. In Grid Algebra, you are shown the arrows when reading and writing expressions and how to move operate, unlike the normal maths lesson.
U101: This has helped us know how algebra is worked out. You are able to see and get to know how algebraic expressions are formed.
About one fifth of all pupils described a sense of apparent ‘ease’ in software-based tasks.
G048: Understanding algebra became easier. Learning by using grid algebra is easy and methods for calculating sums are easier, like inverse journeys for equations.
U100: Easy to learn. It has helped me find another interesting way of learning math thus making me love it.
R228: Using Grid Algebra is easier to understand and it is easy to learn with. It helped me to improve in algebra which would most of the time, prove difficult to me.
The views indicated the pupils’ appreciation of Grid Algebra as a tool ‘to learn’ as well as to ‘learn with’. Visual imagery provided by the software feedback was explicitly mentioned as supportive by 31 pupils.
P171: It has given me another way of learning. It has made me form a mental picture when calculating algebraic expressions, e.g. when I remember how the arrows move in the computer.
U108: When taking my exams, I recall the Grid Algebra lessons that were so scintillating and interesting. The layout is so real and unforgettable thus sticks in my mind, so I imagine a grid in my exam. It helps a lot.
R239: This is because it has really assisted me in carrying out various algebraic questions and improved my understanding of algebra. It improved my learning skills while working algebraic questions since while answering the questions, I had an idea of how the question would look like in Grid Algebra.
These pupils described forming and using mental images of Grid Algebra. The views suggest that dynamic visual imagery can support some pupils’ accessing of mathematical concepts in both classroom learning and examinations. ‘Fading’ of visual support was described by other pupils who felt the scaffolding offered by the software had its limitations.
P207: It helps us visualise the work in algebra but does not help when it comes to exams. The grid algebra lessons helped to understand the steps followed in an equation.
However, this view was not shared by all pupils.
R217: It shows the exact steps to take without taking shortcuts and can be used in exams.
Nevertheless, these views highlight the effect of visual imagery on pupils’ constructions of mathematical meanings, which I will discuss in Chapter 5. The immediate feedback provided by the software was regarded as vital to pupils’ learning; importance was attached to its timing.
G032: It marked the work we did on the spot giving us scores and encouraged us to work some more. In class, it is not usual for the teacher to mark our work on the spot.
R257: Since learning algebra with ICT is interesting, one does not forget since they had a nice time. We get to learn mathematics in a more relaxed way than being in class. Sometimes the teacher bores or criticises you whenever you wrong a question (sic). But in the computer, you’ll be told ‘Incorrect’ and learn afterwards.
It seems that the temporary nature of the software feedback affected these pupils’ self- efficacy when learning.
U078:When we are wrong, the book will just be filled up with red crosses! And every time you open your book, it puts you off maths! With the computer, okay the bins and red crosses appear when you are wrong then disappear, no matter how many sums you get wrong, it does not stay with you. That motivates you!
K133:It made me stop and wonder, I thought I knew this! Then I was like Okay, let me try and get a better result in the next one.
These responses illustrate the pupils’ ability to respond to set-backs: their self-efficacy. They welcomed the instant and non-judgemental nature of the software feedback. It implied pupils’ willingness to learn from their mistakes, as indicated in the MBRQ (see Section 4.3).
Many pupils’ willingness to assume more control of their own mathematical learning through active participation in ICT-enhanced activity was clearly indicated in Figure 4.2. They valued making and seeing connections for themselves rather than being told facts in lessons by their teachers; they laid emphasis on ‘hands-on’ Grid Algebra use, and valued knowing why. Thus, these pupils prescribed a supportive role for mathematics teachers in pupils’ learning.