Results and Implications

Students indicated that they eventually forgot about being seen with the web

camera as when they maximized the screen with the Excel application this window went

to the back. This perhaps improved the observation process as there was less sense of

feeling that they were being watched. This however did not mean that any Hawthorne

effect had been removed completely (see Landsberger, 1958). The Hawthorne effect is

where students may work harder on tasks in response to being observed. The students

also had the convenience of using their own computer and environment, so they were

aware of where applications were located and where they could find implements such as

pen, paper or calculators.

However, some students indicated that they often felt a break in concentration

possibly more explanations) occurred during the practising of the mechanical task and

thus for the rest of the tasks there were few explanations. Most participants eventually

said what they were doing felt like the same thing being repeated for both the tasks

(mainly for the mechanical and interpretive tasks) and the software boxes. For example,

Student 1B said the following when she was doing the tasks in the following sequence

of software boxes

Open-box: “Same calculations … same as the first one” …“They’re all the same” Black-box: “It’s quicker, but all the same”

Glass-box: “Different layouts for them, but all the same”

Student 1B in her reference to open-box software was indicating that the tasks

were all similar. When it came to doing the tasks using the black-box software, she

mentioned that the black-box software was quicker and the glass-box software had a

different layout, but essentially they were all the same when it came to solving the tasks.

The expected values tasks were perhaps quite simplistic as most students made

passing comments to the effect that it would have been faster to do it by hand (for

example Student 2J) and whether they could use pen and paper instead (e.g. Student

6R). Student 5Ch however was the opposite in that initially he did not want to do the

calculations using pen-and-paper as he did not understand how to calculate the expected

values. Afterwards using the different software boxes, he then preferred to solve the

tasks using pen-and-paper as he indicated that he had now learnt how to do the tasks by

watching the software boxes. All students were able to obtain full marks on the

mechanical tasks. Further, as the same instructional materials were used by all students

this meant that there was no privileging of prior teaching styles (Kendal and Stacey,

2001) and therefore this would not influence the way they learnt.

some students opted to use trial-and-exploration methods of testing values either

through the software boxes, or using algebraic equations to solve for the unknown

quantity, r, although r was not required to be solved. This showed that for the

interpretive tasks, students brought in other knowledge such as the algebraic models but

were also likely to resort to the software boxes by exploring and testing their

hypotheses. For example, in the interpretive task, some students worked out that r ≤

90% and r ≥ 20 %, and tested values for r in this range, until they could conclude which

game was better. Also, students appeared to explore with the black-box or the glass-box

software more than the open-box software as the latter required more interactivity.

For the interpretive tasks, there appeared to be more self-explanations occurring

for the black-box and glass-box than for the open-box software. Whilst for the

constructive tasks, the students using the open-box and glass-box software appeared to

have more self-explanations. However, this may be the nature of the tasks (being of a

contextual nature) rather than the software boxes itself. Thus, in the Main Study there

was a need to find out whether it was the task type that was causing the use of the

software boxes in that way, that is, if there was a relationship between the task types

and the software boxes used, or whether it was just the tasks alone that elicit that type of

reaction.

Also, there was a likelihood that open-box software promoted self-explanations

more than the others since it had prompts. The open-box software was also considered

tedious by the students, and perhaps the reason for avoiding it during exploration. The

possible reason for this was that it perhaps needed a higher cognitive effort than the

others, the procedures overly simplistic or it was just badly designed as students often

4.4 Pilot Study 3: Linear Programming

Based on the expected values study, it was noted that students thought the tasks

were quite simple and resorted to using pen-and-paper. This meant that a more

complicated mathematical domain was needed that students could not easily work out

with pen-and-paper in order to determine the influence of the software and steps.

Further reasoning for choosing linear programming was also indicated in Section 3.4.1

(p.66). However, choosing the actual aspect of linear programming was challenging.

The aim of this pilot study was to test whether the linear programming domain could

adequately be used as tasks for testing the three software boxes.

There were three main parts in linear programming: the formulation of the

problem, the solution to the problem and the sensitivity analysis. When it came to steps,

the research focused on the solution part requiring the simplex method as this was part

of the task that could be easily converted into a type of software box, although the

sensitivity analysis could do the same; this would have required more complex linear

programming concepts such as the duality of the problem. The research needed to keep

the introduction of new concepts to the students at a minimum to ensure there was not a

large cognitive effort required. The dual problem required a more complex simplex

method (two phase simplex method) which would be beyond the student to learn or

understand in their first introduction to linear programming.

In using the simplex method to solve linear programming problems, there are

several steps that the user may have to do (Winston, 1994):

1.Convert the problem into canonical form

2.Decide what are the basic variables

5.Decide the pivot row

6.Perform elementary row operations

Whilst all these steps were needed, perhaps the key steps were from Step 3 to

Step 5, as these required some ‘rule of thumb’ to do. For example, at Step 3 deciding

the entering variable indicated which variable should be increased to provide the largest

profit, whilst calculating the ratio and deciding the pivot row indicated how much the

variable can be increased without violating the constraints (see also Section 3.7, p.86).

The formulating of a word problem may or may not add to conceptual and

procedural knowledge but this could not be translated easily to the software boxes,

although perhaps it was able to add context to the students. As such the proposal for the

tasks was worded problems that were already formulated for the students. An added

advantage of using already formulated problems was that this ensured all students

starting from the same point rather than having to account for wrongly formulated

problems. Further, as the formulation would most likely occur through pen-and-paper,

this type of data would be lost or obscured through the remote observation method.

4.4.1 Design of Study

Thus only the simplex algorithm which solved the problem using linear

algebraic methods was considered. These were again developed in Excel spreadsheets

using VBA, because although several linear programming software packages were

examined (e.g. Excel Solver, MathLab, Lindo and MathCad), the software packages

were unable to demonstrate the abilities of the black, glass and open-box software.

Figure 13 represents a schema of the steps required for developing the three software

boxes. Two options were considered for the open-box software (represented by OB in

the figure). In the expected values pilot, the students had to do several arithmetic

do only one operation per step as it minimised the cognitive effort required by the

students. Hence the choice of choosing the appropriate pivot variable was selected as

the step (OB current). The steps for black-box and glass-box software are represented

by BB and GB respectively in the figure.

(1) Enter

Numbers (Iteration2) Click (3) Choose _Variable

(4) Variable correct? No Yes (7) Display Iteration (8) Problem Solved? No (9) Display Problem Solved Yes Steps involved: BB: 1,2,7 GB: 1-2, 7-9 OB (current) : 1-4, 7-9 OB (original) : 1-9 (5) Choose Pivot Row (6) Variable correct? No

Figure 13: The schema for developing the black-box, glass-box and open-box software

Snapshots of the black-box software (Figure 14), the inputting of the values

(Figure 15), the choice of pivot variable for the open-box software (Figure 16) and the

iterations and solutions for both glass-box and open-box software (Figure 17 and Figure

18) are shown below. An annotated screen shot (Figure 7, p.73) along with the

Figure 14: Linear programming black-box software showing the canonical form and solution

Figure 15: Linear programming software-box showing input problem screen

Figure 17: Linear programming software (glass-box or open-box) showing first iteration

Figure 18: Linear programming software box (glass-box or open-box) showing iterations and solution

The study design was again similar to that for the expected values, but in this

that students may acquire through the learning of three different software boxes. Three

problems were given but this time each problem had three parts, where each part was a

mechanical, interpretive and constructive task (see Table 17).

These three tasks were all related to each other. Both the interpretive and

constructive tasks were dependent on the mechanical task. This was used to diminish

the feeling that all the tasks were the same and ensured students would not feel as if

they were repeating the same task again. Students were thus required to compute the

mechanical task correctly in order to do the interpretive and constructive tasks.

Table 17: Example of a linear programming problem

Linear Programming Problem: a) Solve

Max 2x1 + x2

2x1 + x2 ≤ 100 (constraint A) X1 + x2 ≤ 80 (constraint B) X1 ≤ 40(constraint C)

X1, x2 ≥ 0(Mechanical Task: 2 marks)

b) If x1 = no. of toy trains manufactured and x2 refers to the no. of toy soldiers

manufactured, and constraint A refers to painting hours, constraint B to carpentry hours and constraint C, the demand for toy trains. Interpret what this solution means to the toy company who wants to maximize their profit by producing toy trains and toy soldiers. Provide as detail answer as possible. (Interpretive Task: 2 marks)

c) If the cost of trains has increased by £0.50, how would this affect the number of toy trains and toy soldiers being sold? Provide as detail as an answer as possible.

(Constructive Task: 2 marks)

In the expected values study, the students solved all the mechanical tasks

correctly because it only required the students inputting the values and clicking

calculate. Even in the open-box software where students had to interact with steps in the

expected values study, the students were still able to choose the appropriate steps.

all students will calculate the mechanical task correctly. This was particularly true in the

linear programming open-box software, in that, as students only had to choose the

correct pivot variable, this would not affect the computation in any way. There was also

the possibility that the students would get the wrong answer if they inputted values

incorrectly for any of the software boxes. Thus, the researcher ensured that the students

entered the correct values and by doing this, it meant that students were able to achieve

the same performance scores for the mechanical tasks regardless of software boxes.

The three problems were selected; two were of application types and the other

was an abstract type. The first application problem dealt with the manufacturing of toys

and the second with the manufacturing of furniture. Further, the abstract problem and

the application problems were mixed in to determine whether more explanations were

occurring for the application problems versus the abstract problem. However, this

would only be an indication since there was only a comparison between three problems.

Only for the mechanical tasks were students required to use the software boxes.

Students could have answered the constructive task with or without software, that is, it

was possible to solve the constructive task using pen-and-paper. It was expected that for

the interpretive task, that students would not use the software boxes. The software boxes

were used only to show the procedural steps and hence it was expected from examining

the software boxes that students will build their procedural knowledge. If indeed there is

a conceptual-procedural link as suggested by Rittle-Johnson et al. (2001) (see Sections

1.3, p.3 and 2.3.1, p.19) then students’ conceptual knowledge should be impacted on

when they examine the procedural steps.

However, as indicated in Section 2.3.3 (p. )21 , for students to build any

conceptual knowledge, it would depend on the approach (that is their level of

instructional materials. Thus, if students engage with the procedural steps from the

simplex algorithm, they may notice that the slack variables’ values are also calculated.

Conceptually a student may realise that a calculated slack value would mean that there

is a surplus of resources for that constraint. The constructive tasks are devised to take

advantage of these calculated slack values, that is, in all of the constructive tasks, the

students are asked what will occur when a constraint with surplus resources was

increased.

Therefore, if the students were able to engage with the procedural steps and also

build conceptual knowledge, they probably would not need the software box to

recalculate the constructive task but instead determine the answer from examining the

linear programming problem and its calculated answer (from the mechanical task). If

they were unable to build this conceptual knowledge, then recalculation, that is using

the software box, would be their only option. Further a difference in interpretive task

scores for the software boxes may also indicate that students were able to build

conceptual knowledge from the software boxes, as the interpretive tasks mostly require

the application of conceptual knowledge.

Since the interpretive and constructive tasks were dependent on the answer from

the mechanical task, it was imperative that the students got this correct and hence the

researcher ensured that numbers entered were correct. Further, the software boxes were

devised to indicate to the student when the best solution was found. In this Pilot Study,

three students participated to test one software box each (that is Student 1 tested the

black-box, Student 2 the glass-box and Student 3 the open-box). The remote

observation process occurred similarly to that of the expected values study.

In document Students' approaches to mathematical tasks using software as a black-box, glass-box or open-box (Page 126-136)