The Impact of Sequencing and Prior Knowledge on Learning Mathematics Through Spreadsheet Applications

TL;DR: An experiment, using spreadsheets to assist studient learning of mathematics, indicated that for studients with little knowledge of spreadsheets, sequential instruction on spreadsheets followed by mathematics instruction was superior to a concurrent presentation.

Abstract: According to cognitive load theory, instruction needs to be designed in a manner that facilitates the acquisition of knowledge in long-term memory while reducing unnecessary demands on working memory. When technology is used to deliver instruction, the sequence in which students learn to use the technology and learn the relevant subject matter may have cognitive load implications, and should interact with their prior knowledge levels. An experiment, using spreadsheets to assist studient learning of mathematics, indicated that for studients with little knowledge of spreadsheets, sequential instruction on spreadsheets followed by mathematics instruction was superior to a concurrent presentation. The reverse was found for studients with more knowledge of spreadsheets. These results are explained in terms of cognitive load theory.

Summary

CLT

• For the last two decades, CLT has been successfully employed to guide instructional design.
• Cognitive load theorists argue that overloading working memory inhibits learning, and consequently, instructional procedures are most effective when unnecessary cognitive load is kept to a minimum (for a more detailed discussion see Sweller, 2003; Sweller et al., 1998) .

Worked Examples

• Early research into CLT found that instructional formats that required problem-solving search strategies imposed a heavy working memory load that retarded learning (Sweller, 1988) .
• To avoid such problem-solving approaches and enhance learning, instructional designers have successfully employed worked examples in a number of domains, including mathematics (see Cooper & Sweller, 1987; Paas & van Merriënboer, 1994; Sweller & Cooper, 1985) .

Integrating Instruction

• Integrating instruction was a strategy devised to avoid the split-attention effect.
• The split-attention effect typically occurs when instructional materials provide two sources of material, such as a diagram and some explanatory text.
• If both sources are needed to understand the material, the learner is forced to mentally integrate the two sources.
• This process increases cognitive load and is extraneous to schema acquisition and automation.
• From an e-learning perspective, research has shown that the split-source of a computer screen and manual can be eliminated by integrating all material into a computer manual only (Chandler & Sweller, 1996) , or onto a computer screen only (Cerpa, Chandler, & Sweller, 1996; Moreno & Valdez, this issue) .

COGNITIVE LOAD IMPLICATIONS FOR SEQUENCING THE LEARNING OF SPREADSHEET AND MATHEMATICS SKILLS

• Both spreadsheet and mathematical tasks are high in element interactivity and intrinsic cognitive load.
• Learning was enhanced if supportive information was presented sequentially before or during practice.
• Furthermore, van Merriënboer et al. (2003) have argued that sequential order is particularly important in areas of high element interactivity.
• There is an interaction between prior knowledge levels of learners and the amount of information included in instruction.
• The authors hypothesized that students with a low-level knowledge of spreadsheets would learn mathematics more effectively if the relevant spreadsheet skills were learned prior to attempting the mathematical tasks.

METHOD Participants

• Ninth-grade high school students (N = 24) from an independent boy's school in the Sydney metropolitan area participated in the study.
• Some of the students had no experience, whereas others had used spreadsheets to draw graphs, collect data, and complete simple budgets.
• Furthermore, these students had been assessed by their school as being similar in mathematical ability, and had been grouped in the same mathematics class.
• Each participant was allocated to one cell of a 2 × 2 between-subjects factorial design, with instructional format (sequential or concurrent treatment) as the first factor, and perceived spreadsheet ability as the second factor.
• Matching was achieved by using school-based rankings.

Materials

• Materials were designed for instruction (acquisition phase) and testing (test phase).
• The mathematical concepts covered in the experiment required the students to understand the relationships between the calculated values for gradient (m) and y intercepts (C) in the y-inter-cept form (y = mx + C ) of the linear function, and the visual representation of such information in graphical form.
• The second section consisted of the mathematical content to be learned, and employed the use of the spreadsheet skills learned in the first section.
• Both booklets contained the same instructions and activities.
• Students were required to complete four tasks for each lesson, using an Excel spreadsheet by following written instructions.

Procedure

• Prior to starting the study, students were told that they would receive three lessons designed to assist them in developing their spreadsheet skills in the area of linear modeling, and that at the conclusion of the lessons they would complete a short examination containing a mathematics and computer skills section.
• The acquisition phase consisted of three self-contained lessons, which were delivered on separate school days.
• While some participants moved through certain lessons more quickly, all participants completed the activities in the required time.
• Each booklet was fully self-contained and designed for use by a novice, requiring no explanation from the teacher, with the exception of a reminder " to read everything very carefully" and a check to ensure that all participants knew how to save their work into their own personal folder.
• At the conclusion of the test phase students were required to rate how easy or difficult to understand they found the instructions.

RESULTS

• Because of the extended nature of the experiment (four lessons), absenteeism led to four students not completing all tasks.
• These students were excluded from the study, leaving 9 students in the sequential group and 11 in the concurrent group.
• Because of the small sample size and the initial assessment that all students were of similar mathematical ability, the 4 matched partners of the excluded students were not excluded.
• Mean scores for each group, on both mathematical and spreadsheet tasks in the test phase, as well as the self-rating cognitive load measures, were recorded (see Table 2 ).
• These scores were analyzed using 2 × 2 analysis of variance with instructional group (sequential or concurrent) and spreadsheet knowledge (high or low) as the independent variables.

Performance Scores

• Simple effects tests indicated that the less experienced spreadsheet group scored higher on the math test if they received sequential instruction compared with a concurrent format: t(6) = 2.57, p = 0.04.
• For the more experienced spreadsheet group there was no significant difference found.

DISCUSSION

• The main hypothesis tested in this study was supported: Students with a low-level knowledge of spreadsheets learned mathematics more effectively if the relevant spreadsheet skills were learned prior to attempting the mathematical tasks.
• These results can be explained in CLT terms.
• The possibility that experienced spreadsheet learners may benefit from a concurrent approach should be a target for future research.
• In addition to sample size, another potential weakness of this study was that subjective measures of cognitive load were collected after the final test was completed.
• Too frequently, it is assumed that such familiarity can occur simultaneously while learning a subject discipline through e-learning.

Figures

Table 1 Instructional order for each group per lesson.

Table 2 Group means and standard deviations (in brackets) of test scores and cognitive load measures

Full text

The Impact of Sequencing and Prior
Knowledge on Learning Mathematics Through
Spreadsheet Applications
Tracey Clarke
Paul Ayres
John Sweller
According to cognitive load theory,
instruction needs to be designed in a manner
that facilitates the acquisition of knowledge in
long-term memory while reducing
unnecessary demands on working memory.
When technology is used to deliver
instruction, the sequence in which students
learn to use the technology and learn the
relevant subject matter may have cognitive
load implications, and should interact with
their prior knowledge levels. An experiment,
using spreadsheets to assist student learning
of mathematics, indicated that for students
with little knowledge of spreadsheets,
sequential instruction on spreadsheets
followed by mathematics instruction was
superior to a concurrent presentation. The
reverse was found for students with more
knowledge of spreadsheets. These results are
explained in terms of cognitive load theory.
Using new technologies to enhance the learn-
ing and teaching of mathematics is highly rec-
ommended by many professional teaching
associations. However, when learning from
computer-based instructional material, a num-
ber of cognitive load theory (CLT) principles
need to be followed to ensure that learning is
maximized. While much is known about how
computer-based materials should be presented
to avoid negative effects such as split-attention
and redundancy (e.g., see Sweller, van
Merriënboer, & Paas, 1998), less is known about
the interactions associated with learning how to
use technology while simultaneously learning
mathematical concepts. How should instruc-
tional materials be structured so learners can
employ technology in order to enhance under-
standing of mathematics? In this article we
investigate how sequencing the learning of
spreadsheet skills affects learning mathematics.
CLT
For the last two decades, CLT has been success-
fully employed to guide instructional design. A
basic assumption of CLT is that interactions
between working memory and long-term mem-
ory play a significant role in learning. When
dealing with novel information, working mem-
ory is extremely limited. Humans are only able
to store and process a few novel combinations of
elements or chunks at any given time (Miller,
1956). In isolation, such a restriction might sug-
gest that humans are incapable of processing
complex materials. Long-term memory, by
ETR&D, Vol. 53, No. 3, 2005, pp. 15–24 ISSN 1042–1629 15
AAH GRAPHICS, INC. / (540) 933-6210 / FAX 933-6523 / 06-23-2005 / 17:55

effectively altering the capacity and duration
limits of working memory, permits complex
processing. In contrast to working memory,
humans can store vast quantities of information
in long-term memory. In particular, it is argued
by cognitive load theorists (see Paas, Renkl, &
Sweller, 2003; Sweller, 2003; Sweller et al., 1998)
that information stored in long-term memory
can vastly increase the capacity of working
memory. The limitations of working memory
only apply to novel combinations of elements.
Working memory has no known limitations
when dealing with previously learned informa-
tion stored in long-term memory (Ericsson &
Kintsch, 1995).
Information is stored in long-term memory in
the form of schemas. A schema (see Chi, Glaser,
& Rees, 1982; Larkin, McDermott, Simon, &
Simon, 1980) can be defined as a cognitive con-
struct that allows multiple elements of informa-
tion to be treated as a single element categorized
according to its use. When such schemas are
brought into working memory they can be
treated as a single element rather than many,
freeing up working memory resources, and
reducing overall cognitive load. Automation
(Kotovsky, Hayes, & Simon, 1985) has a similar
function: It also reduces cognitive load by reduc-
ing the burden on working memory when infor-
mation can be processed automatically without
conscious processing. The possession and auto-
mation of schemas enable humans to engage in
complex activities in spite of a very restricted
working memory. CLT proponents assume that
students find many tasks difficult to learn
because of the limitations of human working
memory when dealing with novel material, and
that problem-solving skill develops by the con-
struction of domain-specific schemas in long-
term memory (see Kalyuga, Ayres, Chandler &
Sweller, 2003; van Merriënboer & Ayres, this
issue).
Cognitive load theorists argue that overload-
ing working memory inhibits learning, and con-
sequently, instructional procedures are most
effective when unnecessary cognitive load is
kept to a minimum (for a more detailed discus-
sion see Sweller, 2003; Sweller et al., 1998). CLT
researchers have identified three sources of cog-
nitive load during instruction: Intrinsic, extrane-
ous and germane cognitive load (see Paas, Renkl
et al., 2003; Sweller et al., 1998). Intrinsic cogni-
tive load is load placed on working memory by
the intrinsic nature of the materials to be learnt.
Extraneous cognitive load is the load placed on
working memory by the instructional design
itself and germane cognitive load is the load
evoked by the instructional materials that assist
the process of schema formation. Whereas ger-
mane cognitive load is considered positive
because working memory resources are directly
focused on learning, the other two forms of cog-
nitive load can seriously impede learning.
CLT researchers have identified strategies to
reduce intrinsic cognitive load, such as isolating
integrated elements in instructional materials
(Pollock, Chandler, & Sweller, 2002) and simple-
to-complex sequencing of learning tasks (van
Merriënboer, Kirschner, & Kester, 2003). How-
ever, much of the research has focused on devis-
ing design strategies to reduce extraneous
cognitive load (for a summary see Sweller et al.,
1998; Sweller, 1999). The use of worked exam-
ples and physically integrating disparate
sources of information are two such successful
strategies particularly relevant to this study, and
are, therefore, discussed in more detail here.
Worked Examples
Early research into CLT found that instructional
formats that required problem-solving search
strategies imposed a heavy working memory
load that retarded learning (Sweller, 1988). To
avoid such problem-solving approaches and
enhance learning, instructional designers have
successfully employed worked examples in a
number of domains, including mathematics (see
Cooper & Sweller, 1987; Paas & van Merriën-
boer, 1994; Sweller & Cooper, 1985). A worked-
example approach requires learners to study
solutions to problems rather than just solve them,
although in most applications of the worked-
examples approach, learners also solve some
problems, either in a paired format (study one,
solve a similar one, see Cooper & Sweller, 1987),
or as completion problems (completion of a par-
tial solution, see Paas & van Merriënboer, 1994).
AAH GRAPHICS, INC. / (540) 933-6210 / FAX 933-6523 / 06-23-2005 / 17:55
16 ETR&D, Vol. 53, No. 3

Integrating Instruction
Integrating instruction was a strategy devised to
avoid the split-attention effect. The split-attention
effect typically occurs when instructional mate-
rials provide two sources of material, such as a
diagram and some explanatory text. If both
sources are needed to understand the material,
the learner is forced to mentally integrate the
two sources. This process increases cognitive
load and is extraneous to schema acquisition
and automation. Researchers have overcome
this impediment to learning by devising strate-
gies that physically integrate the two sources,
thus eliminating split attention (see Tarmizi &
Sweller, 1988). From an e-learning perspective,
research has shown that the split-source of a
computer screen and manual can be eliminated
by integrating all material into a computer man-
ual only (Chandler & Sweller, 1996), or onto a
computer screen only (Cerpa, Chandler, &
Sweller, 1996; Moreno & Valdez, this issue).
COGNITIVE LOAD IMPLICATIONS
FOR SEQUENCING THE
LEARNING OF SPREADSHEET AND
MATHEMATICS SKILLS
According to CLT, to use spreadsheets effec-
tively to learn mathematics, some principles
need to be followed in the design of the instruc-
tional materials. If specific spreadsheet skills
need to be learned first in order to be useful in
learning mathematics, then sequencing order is
critical. Both spreadsheet and mathematical
tasks are high in element interactivity and intrin-
sic cognitive load. If both learning tasks are pre-
sented simultaneously, it is likely that cognitive
load will be compounded. As a consequence,
learning may be inhibited on both tasks; there-
fore, to maximize learning in the mathematical
domain, spreadsheet skills should be mastered
and consolidated first.
Recent work on presentation sequencing pro-
vides support for this argument. Kester, Kirsch-
ner, and van Merriënboer (2004; in press) and
Kester, Kirschner, van Merriënboer and Baumer
(2001) found that the simultaneous presentation
of all necessary information, either before or
during practice, had no beneficial effect on
learning. However, learning was enhanced if
supportive information was presented sequen-
tially before or during practice. Furthermore,
van Merriënboer et al. (2003) have argued that
sequential order is particularly important in
areas of high element interactivity. When using
spreadsheets to learn mathematics, if students
are able to learn the “ necessary” (p. 9) spread-
sheet skills first, schemas will be acquired in
long-term memory which can be activated dur-
ing application of those skills to learning new
mathematical concepts. Activation of such sche-
mas will reduce cognitive load when learning
the new information. Tasks in mathematics and
spreadsheets are both high in terms of element
interactivity, so this argument is consistent with
the conditions met in this study.
However, recent research has found that the
effectiveness of instructional materials is depen-
dent on the expertise of the learner. There is an
interaction between prior knowledge levels of
learners and the amount of information
included in instruction. In some circumstances,
information that is essential for a novice has
been found to be redundant for those with more
expertise. This interaction is called the expertise
reversal effect because, with increasing levels of
expertise, strategies that are effective for novices
have been shown to be ineffective for more
knowledgable students (Kalyuga et al., 2003). As
a rule, inexperienced learners need much more
guidance than more experienced learners in any
particular domain (Mayer, 2001; Renkl & Atkin-
son, 2003). Worked examples, for instance, are
very effective for novices (van Merriënboer and
Ayres, this issue), but as expertise develops
(resulting in more sophisticated schemas), a
problem-solving approach may be superior
(Kalyuga, Chandler, & Sweller, 2001).
Because of the influence of expertise,
sequencing the acquisition of spreadsheet skills
prior to mathematical acquisition may not be
necessary. Students experienced in using
spreadsheets may possess schemas that will pre-
vent cognitive overload when simultaneously
combining technology and mathematical tasks.
In such a case, students may benefit from having
materials structured in a concurrent format
because of a reduction in temporal split atten-
tion (Mayer & Anderson, 1991; 1992). A concur-
AAH GRAPHICS, INC. / (540) 933-6210 / FAX 933-6523 / 06-23-2005 / 17:55
SEQUENCING, PRIOR KNOWLEDGE, AND SPREADSHEETS 17

rent format and its attendant reduction in split
attention and cognitive load may enable stu-
dents to focus on the relations between
spreadsheet knowledge and mathematics.
This study, based on CLT, investigated the
timing of learning spreadsheet skills in applying
spreadsheet applications to mathematics. We
hypothesized that students with a low-level
knowledge of spreadsheets would learn mathe-
matics more effectively if the relevant
spreadsheet skills were learned prior to attempt-
ing the mathematical tasks. We also hypothe-
sized that students experienced in spreadsheets
would not need such a sequenced approach,
being able to benefit from a more integrated
approach, where new spreadsheet skills and
mathematical concepts were learned together.
To test these hypotheses, students were assigned
to one of two instructional formats. The sequen-
tial group was given instructions on
spreadsheets prior to applying this knowledge
to learning mathematics. The concurrent group
was given the spreadsheet skills and mathemat-
ical concepts in an integrated format. To test the
effects of expertise, students with differing lev-
els of experience with spreadsheets were
assigned to each instructional group.
The primary goal of the study was to investi-
gate how the given conditions would affect learn-
ing mathematics. However, a secondary goal was
to investigate how these conditions would also
affect learning spreadsheet skills. Consequently,
both mathematics and spreadsheet skills were
tested and included as dependent variables. Fur-
thermore, a subjective measure of how difficult
participants found the instruction was collected
and included in the analysis.
METHOD
Participants
Ninth-grade high school students (N = 24) from
an independent boy’s school in the Sydney met-
ropolitan area (Australia) participated in the
study. Some of the students had no experience,
whereas others had used spreadsheets to draw
graphs, collect data, and complete simple bud-
gets. Based on class tests, students had been
assessed by the school as above average in gen-
eral mathematics ability, and were following a
mathematics curriculum specifically designed
for students in the top 25% of the state of New
South Wales. Furthermore, these students had
been assessed by their school as being similar in
mathematical ability, and had been grouped in
the same mathematics class.
Each participant was allocated to one cell of a
2 × 2 between-subjects factorial design, with
instructional format (sequential or concurrent
treatment) as the first factor, and perceived
spreadsheet ability as the second factor. To con-
trol for possible mathematical effects, students
were matched in pairs on mathematical ability,
and assigned to one of the treatment groups at
random (both n = 12). Matching was achieved by
using school-based rankings. It should be noted
that this matching procedure was considered an
extra precaution, because the whole class were
approximately of the same mathematical stan-
dard. To ascertain spreadsheet ability, students
were asked to rate their current spreadsheet
ability according to a 4-point scale of advanced,
intermediate, beginner, and no experience. These
ratings were used to classify students as high
spreadsheet experience (advanced or intermediate
selections; n = 14) or low spreadsheet experience
(beginner or no experience; n = 10).
Materials
Materials were designed for instruction (acquisi-
tion phase) and testing (test phase). The mathe-
matical purpose of the experiment was to use
spreadsheets to assist the development of
understanding of graphical representations of
linear functions when they are given in both
table and equation form. Prior to commencing
the study, students were halfway through an
algebraic module on coordinate geometry. Stu-
dents had used formulas for finding the gradi-
ents, midpoints and distances to given sets of
coordinate points, but had not applied this
knowledge to graphical representations. The
mathematical concepts covered in the experi-
ment required the students to understand the
relationships between the calculated values for
gradient (m) and y intercepts (C) in the y-inter-
AAH GRAPHICS, INC. / (540) 933-6210 / FAX 933-6523 / 06-23-2005 / 17:55
18 ETR&D, Vol. 53, No. 3

cept form (y = mx + C ) of the linear function,
and the visual representation of such informa-
tion in graphical form. Consequently, at the con-
clusion of the instructional period, the students
should have been able to graph various forms of
the linear equation, and reverse the process by
calculating the linear formula from graphical
representations. To create a graph (chart) of an
equation in Excel™ , the user must first create a
table of values to be graphed. This process rein-
forces the fact that an equation is simply a rela-
tionship that connects independent and
dependent variables, and that all three forms—
(a) equations, (b) tables of values, and (c)
graphs— of a linear model are alternate repre-
sentations of each other.
Acquisition phase. Instruction booklets were
specifically designed to teach the required
spreadsheet skills (entering formulas, using the
function facility, and creating line graphs) and
mathematical concepts. Cognitive load princi-
ples were used to integrate text and screen cap-
tures (see Figure 1) to create an effective learning
environment and reduce split attention. Total
self-containment of instruction was achieved by
the extensive use of screen captures. Each book-
let was designed to cater for three 50-min les-
sons.
The two instructional format groups received
different booklets. For the sequential group, the
instruction booklet was divided into two sec-
tions. The first section contained instructions to
develop all of the necessary spreadsheet skills
(similar to a how-to computer manual). The sec-
ond section consisted of the mathematical con-
tent to be learned, and employed the use of the
spreadsheet skills learned in the first section.
Hence the mathematical concepts were learned
after the development of the spreadsheet skills.
The instruction booklet given to the concurrent
group combined spreadsheet skill development
and mathematics instructions, developing the
spreadsheet skills and their application to math-
ematics concurrently. Table 1 summarizes the
particular mathematical and spreadsheet con-
cepts learned and the order in which they were
presented over the three lessons. The order in
which spreadsheet skills and mathematical con-
cepts were introduced was identical for both
groups with one exception. For the concurrent
group, the spreadsheet skill “ gradient and slope
functions” was introduced at a later point to
coincide with its mathematical applications.
Figure 1 An example of the integration of instructional texts with screen captures.
AAH GRAPHICS, INC. / (540) 933-6210 / FAX 933-6523 / 06-23-2005 / 17:55
SEQUENCING, PRIOR KNOWLEDGE, AND SPREADSHEETS 19