Explanations: Problems, Tasks and Boxes

Whilst a Sequence effect is found, when looking at the number of explanations

across Problems, Tasks and software Boxes, one should note that students were

randomly assigned to Sequence. Each software Box had a similar number of students in

each Sequence which meant that any effect from Sequence will be evened out across

Problems, Tasks and Software Boxes.

Amongst the three problems, students made slightly more explanations for

Problem 2 (64 explanations) than Problem 1 (46 explanations) and Problem 3 (50

explanations). The abstract problem, Problem 3, had a very low percentage of real-life

explanations (8%) compared to the two application problems, Problem 1 (39%) and

Problem 2 (50%). However, Problem 3 had a high percentage of mathematical

explanations (58%) compared to Problem 1 (21%) and Problem 2 (34%). Percentages

are calculated based on the possible number of real-life or mathematical explanations

for each problem (i.e. total number of possible real-life explanations per problem: 38

students × 2 tasks per problem × 1 real-life explanation = 76 real-life explanations).

Figure 26 presents these results.

0 10 20 30 40 50

Problem 1 Problem 2 Problem 3

Problems N um be r of E x p la na ti ons Maths Real

These results suggest that if there is a real-life application task, students were

more likely to rely on making real-life type explanations and to a lesser extent

mathematical explanations to support their answer. However if the task is abstract then

students probably did not see the relation to a real-life situation and rely mostly on

mathematical explanations (Section 6.4.2, p.203 With the application problems, ).

students probably brought in real-life knowledge to help understand the task (Boaler,

1993), whether this hampered their performance or understanding is ascertained in the

next section and Chapter 6.

Overall there were more explanations for constructive tasks (93) than

interpretive tasks (67). Constructive tasks had more than 1.5 times the number of

mathematical explanations (47%) than the interpretive task (28%). The bracketed

percentages are based on the possible number of mathematical explanations per task

(i.e. 114). Students made almost equal numbers of real-life explanations between both

tasks (see Table 29).

Table 29: Number of real-life and mathematical Explanations for Problem and Task

Tasks Interpretive Constructive

Explanations Maths Real Maths Real

Problem 1 5 (13%) 5 (13%) 11 (29%) 25 (66%) Problem 2 8 (21%) 26 (68%) 18 (47%) 12 (32%) Problem 3 19 (50%) 4 (11%) 25 (66%) 2 (5%) All 32 (28%) 35 (31%) 54 (47%) 39 (34%)

Given that students made more mathematical explanations in Problem 3, this

was reflected in the high number of mathematical explanations in the interpretive (50%)

and constructive tasks (66%) and the low number of real-life explanations in both of

these tasks for this problem, as seen in Table 29. The percentages are based on the total

possible number of explanations, per task per problem which is 38. Therefore the

making that type of explanation. Students gave more real-life explanations for Problem

2’s interpretive task (68%) than any other interpretive task whilst Problem 1’s

constructive task attracted the most real-life explanations (66%) amongst all the

constructive tasks. The real-life explanations were the predominant explanation type for

these two tasks (i.e. Problem 1’s constructive task and Problem 2’s interpretive task).

Problem 1’s interpretive tasks and Problem 2’s constructive task had similar number of

mathematical and real-life explanations.

Now whilst all of this is interesting, what is important for this thesis is to know

whether the software Boxes influenced the number and type of explanations. Using a

chi-square test, only marginal significance (χ2(2) = 5.19, p = 0.07) was found such that

the software Boxes were associated with the number of real-life explanations (see

Figure 27). Students using the glass-box software (41%) were more likely to have a

higher number of real-life explanations than those students on the black-box (32%) or

open-box (24%) software. 0 5 10 15 20 25 30 35

Black-Box Glass-Box Open-Box

Software Boxes N um be r of E x p la na ti ons Maths Real

Figure 27: Number of real-life and mathematical Explanations for the software Boxes

This association of real-life Explanations and software Boxes was mostly due to

the students with higher Mathematics Confidence (χ2(2) = 7.65, p = 0.02). Higher

explanations for half of their tasks (constructive and interpretive combined), whilst

higher mathematics confidence students in the open-box software barely gave any real-

life explanations (17%). Higher mathematics confidence students using the black-box

software provided real-life explanations for 30% of their tasks.

Perhaps the open-box software because of its mathematical steps made students

feel that any explanations should be more mathematical in nature and hence they

reduced their number of real-life explanations. The reason why the higher mathematics

confidence students using the glass-box were providing more real-life explanations is

uncertain. However, as the higher mathematics confidence students did not explore with

the glass-box greatly, they probably tried to explain their given answers by providing

real-life explanations. The ratio of mathematical to real-life explanations for all students

was found to be the lowest in the glass-box software (0.75) and the highest in the open-

box software (1.71). Black-box software had a ratio of 1.32.

Thus to sum up this section, students solving the constructive task gave the most

number of explanations. The constructive task asked students to explain what will

happen if a value changes (such as profit or number of products) and this was followed

with why the change (or no change) occurred. This probably gave students a wider berth

to use a range of explanations. Mathematical explanations seemed to be more popular

for Problem 3, for both its interpretive and constructive tasks. The number of real-life

explanations was popular in one interpretive task (Problem 2) and one constructive task

(Problem 1) which were both related to application problems (Problems 1 and 2).

Finally, higher mathematics confidence students were more likely to provide real-life

explanations if they were using the glass-box software.