Grid Algebra tasks

In this section, I focus on describing eight software-generated tasks which I have linked to specific curriculum objectives listed in Section 1.2.4.

1.5.3.1 Task 1: ‘Calculating’

Task 1 is a number activity with tasks at ten levels of difficulty determined by the number of mathematical operations (ranging from 1 to 10) contained in each numeric puzzle.

Figure 1.4: Screenshot of the software-generated task 1: ‘Calculating’

Every exercise has a series of 20 questions, each involving one numeric expression in a ‘blue’ cell within a visible grid as shown in Figure 1.4. Questions appear as they would in printed mathematics textbooks. Pupils were expected to work out each question and then drag their answers from the ‘Number Box’ to the ‘blue’ cell. The software marked predicted responses and provided immediate feedback. Feedback appeared as a score followed by next question, or as a ‘No Entry’ sign in the ‘blue’ cell. Faulty responses hovered below the question. The feedback seemed to enable pupils to pause and reflect on reasoning underlying the responses offered. Further click of the mouse ‘binned’ the user’s response, the score remained unaltered and the next question appeared. After the tenth question, the grid disappeared. Subsequent questions appeared in a single, enlarged cell. The software displayed final results on a score card with a brief comment, and gave the user the option of either continuing with the task by attempting puzzles at a lower, the same or next level of difficulty, or quitting the task

altogether.

This task linked the learning of algebra with arithmetic (Van Amerom, 2003): it required the pupils to make connections between their current and previous mathematical knowledge (McGowen and Tall, 2010; Askew et al, 1997; UK Department for Education, 2007).

1.5.3.2 Task 7: ‘Find the journey (letters)’

This activity consists of ten levels of difficulty describing ‘journeys’, each with between two and eleven mathematical operations, and 10 puzzles on each difficulty level.

Figure 1.5: Screenshot of the software- generated task 7: ‘Find the journey (letters)’

The task for the user is to create a given algebraic expression consisting of both numbers and letters by clicking on the correct cells to show the path traced by a letter across the grid. The ‘journey’ starts from the cell with the letter and ends at the expression. A red cross, ‘X’, appears briefly when the user clicks on an unpermitted cell; this encourages the user to reflect further on the mathematical operation required to recreate the given ‘journey’. The software displays the path traced boldly across the grid as shown in Figure 1.5. It provides options for the user to either start the journey again or indicate they had finished the journey, whereupon instant feedback is offered and the next puzzle appears.

I hoped engagement with this task would enable the pupils to make connections between the physical movements on the grid and mathematical operations. It provided the opportunity to link the dynamic visual representation to formal symbolic algebra for pupils (Clausen-May, 2008; Noss et al, 1997) whilst they mastered using the ICT tool.

1.5.3.3 Task 25: ‘What is the expression?’

A path traced by a letter across the grid from a ‘blue’ cell is given to the user. An expression is required to be typed into an ‘Expression Calculator’ as shown in Figure 1.6.

Figure 1.6: Screenshot of software-generated task 25: ‘What is the expression?’

The user performs between one and ten mathematical operations on a letter represented as expressions in formal algebraic notation which the user is allowed to ‘clear’ or ‘undo’. Each of the ten levels of difficulty consists of ten puzzles. The software provides feedback once the user presses ‘Enter’; it indicates either, “Well done, Correct”, or a ‘No Entry’ sign appears in the calculator followed by, “Incorrect, Ready for next task?”, with no option to re-do the task. The user is allowed time to reflect on these responses before resuming the activity. After the tenth puzzle, the software provides results in the form of a score, a brief comment, and options to attempt the task at a lower, the same or next level of difficulty or to quit the task. This task considered pupils’ new knowledge of the software as a supportive ‘met-before’. Its selection in this study was to educate pupils’ awareness of the use of symbols and letters in algebraic statements, and hence to develop their ‘symbol sense’ (Arcavi, 1994). I hoped that engagement with this task would gauge the pupils’ use of brackets for multiplication and the line notation for division as the accepted symbol convention in algebra (Bell, 1996). This is in line with the specific objective of writing statements in algebraic form (KIE, 2002).

1.5.3.4 Task 13: ‘Make the expression (letters)’ This is a timed activity.

Figure 1.7: Screenshot of software-generated task 13: ‘Make the expression (letters)’

The software provides a ‘clock’ with a time limit in the top left hand corner of the computer screen, an expression below the clock, and a letter, as shown in Figure 1.7. Each difficulty level consists of ten puzzles, each with an expression created by performing between two and eleven mathematical operations on the letter. The user is required to drag the letter across the grid to create the prescribed expression within a given time frame in order to score a point, or to “give up”. They had 10 seconds for level 1, 20 seconds at level 2, 25 seconds at level 3, increasing to 60 seconds at level 10.

This task tested pupils’ knowledge of the working of the software; they read and created the given expressions whilst gauging their acceptance of the formal algebraic notation. This task was aligned with the learning objective of secondary school pupils performing mathematical operations and manipulations with confidence, speed and accuracy (KIE, 2002).

1.5.3.5 Task 12: ‘Inverse journey’

This task has three levels of difficulty: level 1 (2 to 4 operations), level 2 (3 to 5 operations) and level 3 (6 to 7 operations), with puzzles consisting of 20 questions each.

Figure 1.8: Screenshot of software-generated task 12: ‘Inverse Journey’

The screen shows the grid with a letter p in a ‘red’ cell and the ‘journey’ traced by letter p is represented as an algebraic expression in the ‘blue’ cell. Also showing is the ‘Expression Calculator’ with a letter t on its screen and a ‘Magnifier’ in which the algebraic expression (the ‘journey’ traced by the letter p) is equated to the letter t. Each puzzle in this task required the user to type into the calculator the inverse route of the original journey shown, starting from letter t and ending with letter p. Pupils had to think of reversing each mathematical operation in order to ‘undo’ the expression, thereby leaving the letter p on its own. Once the user presses ‘Enter’ to indicate completion of the task, their typed response is transferred to the ‘red’ cell. The expected solution is displayed as shown in Figure 1.8. The software provides scaffolding for the first 10 puzzles: it allows users to see the original journey by clicking a button reading ‘Show original journey’. Each correct inverse journey earns the user one point. A ‘No Entry’ sign indicates the entry of an error; the response remains in the calculator. In this way, the users have a chance to see and reflect upon the algebraic expressions representing the original journey, the inverse journey and the user’s response. I stated in Section 1.5.1that the consequential feedback provided by the software was considered crucial in enabling the pupils to make sense of algebraic meaning (see Section

2.5.4). I hoped that the focus in this task on mathematical processes would endorse a ‘structural’ conception of algebra (Kieran, 1992; Sfard, 1991). The activity stressed the argument by Bell (1996) of the role played by symbolic language and its manipulation in algebraic thinking.

1.5.3.6 Task 21: ‘Simplify’

This is an activity comprising of tasks at ten levels of difficulty, determined by between 2 and 11 mathematical operations in each puzzle, and accompanied by a set of instructions.

Figure 1.9: Screenshot of the software-generated task 21: ‘Simplify’

Every puzzle in a level has a series of 10 questions. Each question is an expression containing both numerals and letters, in a ‘blue’ cell with a letter in the visible grid. Questions appear as they would in the regular mathematics textbooks. The user is meant to type into the

‘Expression Calculator’ provided a simpler expression which is equivalent to the expression given. The software awards one point for a simpler expression, and two points for the simplest, as a way of encouraging the user to think of, and provide their responses in the simplest form. The grid disappears after the fourth puzzle; it leaves the letter and question lingering on the screen. In the eighth question, both the letter and ‘blue’ cell disappear: the question appears in an enlarged cell. When the user types an incorrect expression, a ‘No

Entry’ sign appears in the bottom left part of the calculator screen; the software remarks, “Sorry. That does not get you to the final cell- so 0 points!” The software displays results as shown in Figure 1.9; it gives options for the user to continue with the task by solving puzzles at a lower, the same or next level of difficulty, or quitting the task altogether.

In Section 1.2.3, removal of brackets and simplifying algebraic expressions were two specific objectives to be met by pupils (KIE, 2002). The task presented an opportunity for developing a key 21st Century skill (DEAG, 2013; Luckin et al, 2012; Livingstone, 2012). The pupils were required to synthesise their grasp of various concepts within algebra and arithmetic. It evoked a connectionist belief orientation of teaching mathematics (Askew et al, 1997), which stressed making links within different mathematical concepts, a view shared by Rudduck et al (1994).

1.5.3.7 Task 22: ‘Substitution’

This activity has tasks in 10 levels of difficulty; each level has 20 questions consisting of algebraic expressions with between one and ten mathematical operations. A ‘red’ cell with the expression, and a ‘blue’ cell with a letter whose value is given, are displayed as in Figure 1.10.

The user is required to work out the value of the expression by substituting the letter in the expression, and drag their solution from the ‘Number Box’ into the ‘red’ cell. After the tenth question, the grid disappears. The software provides immediate feedback in the form of either a score and the appearance of the next question. Otherwise, a ‘No Entry’ appears in the ‘red’ cell as well as a ‘bin’; the user’s response lingers on the screen.

The specific learning objective to be met by the pupils was to evaluate expressions by substituting numerical values (KIE, 2002). Success depended on the pupils’ knowledge of multiplication tables as a supportive ‘met-before’. The learning activity aimed to generate reflection through ‘formative feedback’ provided by the software, peers and the teacher (Juwah et al, 2004).

1.5.3.8 Task 6: ‘Expanding and factorising’

This activity involves the use of brackets in tasks presented in five levels of difficulty: Expanding (multiplying), Factorising (multiplying), Expanding (dividing), Factorising (dividing), and a mix of levels 1 to 4. The software displays images as shown in Figure 1.11.

Figure 1.11: Screenshot of software-generated task 6: ‘Expanding and Factorising’

A letter in a cell on the screen is the starting point of two possible routes across the grid. The user is informed that the expression in the green cell represents the ‘journey’ traced by the

letter on the thin green route. The user’s task is to generate the expression for the thick red route. There are 20 questions for each exercise. Later in the task, first the grid disappears after the sixth puzzle, leaving the letter, the expression and the two routes on the screen; after the twelfth puzzle, the routes are hidden, leaving the expression in an enlarged cell attached to the calculator. At this stage, questions appear as they would in mathematics textbooks. Once the user presses ‘Enter’, the software provides feedback as “Correct- 1 point!”, or a ‘No Entry’ sign in the calculator, and “Incorrect. Have a look at the correct solution”; it then gives the result which shows the expressions from the two routes as equal to each other.

KIE (2002) emphasised the introduction and use of new mathematical terminology to pupils. Several researchers (Yerushalmy and Naftaliev, 2011; Clausen-May, 2008; Noss et al, 1997) argued for opportunities for increased ‘connectivity’ for pupils in mathematics. The dynamic visualisation and formal symbolism of the software converged with the activity-based

dialogue as pupils appropriated the emerging algebraic knowledge.

I hoped that including ICT-enhanced activity in the secondary mathematics lessons would help to consolidate the pupils’ conceptual understanding through the provision of alternative pathways to their learning (Abbot et al, 2009). The tasks presented learning content in ways that allowed pupils to build strong connections between the mathematical knowledge they already possessed and what they were learning.