Strategies for equipping students with the necessary skills and knowledge to successfully solve
problems
Dasha Bunks
Liberty University
PSYC 775 - Teaching of Psychology
Dr. Winn
2022
Strategies for equipping students with the necessary skills and knowledge to successfully solve
problems
If you teach mathematics, statistics, or science you probably know the following
situation very well. You have introduced a theorem or physics law in your classroom. When the
students then work on corresponding problems, many of them mainly try to find the correct
numerical solution by applying the corresponding formulae; they do not further reflect on the
rationale of the solution or on the underlying mathematical or physical structures. In this way,
many students become rather proficient in applying formulae.
This is at least true as long as it is clear from the classroom context that the problems to
be solved are related to the relevant formulae and as long as the formulae can be directly
applied without modifications. However, when the students encounter related problems in an
unfamiliar context or embedded in a new type of cover story, many of them are not able to
identify the underlying mathematical or physical structures and to apply the correct theorem or
physics law. Many students also fail when they have to slightly modify a known solution
procedure. Such phenomena show that mechanically applying formulae that were most
recently introduced in the classroom does not deepen understanding and does not enable the
students to apply their knowledge in new contexts or for new, but related problem types.
One reason why many students mechanically apply formulae when solving problems is
that they have not yet developed a sufficient understanding of a theorem or of a physics law
before solving problems. Hence, they simply are not able to solve the problems in a meaningful
way, that is, by considering mathematical or physical structures and corresponding theorems or
laws. One tried-and-tested method to address this problem is to use learning from worked
examples for the initial acquisition of cognitive skills (Sweller, Ayres & Kalyuga, 2011; Paas &
van Gog, 2006; Renkl, in press-b). In this method, the learners study several worked examples
after some principles (e.g., theorems or laws) have been introduced. The students are best
guided to explain the rationale of the worked solutions to themselves. Only after the learners
have developed the understanding that is necessary for meaningful problem solving, they work
on problems.
This short characterization of learning from worked examples illustrates the basic idea
of this learning method. When I introduce this rationale to teachers I frequently hear three
objections. Although these objections all have a “true core,” none of them are sufficient for an
instructor to refrain from using this technique. 119 Objection 1: It’s just for maths and related
stuff. Actually, much on the initial research on learning from worked examples was done with
mathematics and science contents. However, it has been shown that learning from worked
examples can be applied in many different domains, for example, learning to argue (Schworm &
Renkl, 2007); learning about designer styles (Rourke & Sweller, 2009); learning to apply learning
strategies (Hilbert & Renkl, 2009). Objection 2: Worked examples are typical of old-fashioned
pedagogy focusing on algorithms and "recipes".
Yes, worked examples are often used in a way that does not deepen understanding, that
is, mainly as a tool to teach algorithms or simple recipes. In many of these cases, just one
worked example is presented. Note, however, that learning from worked examples, as
conceptualized by their proponents, means that a series of examples are presented with the
main goal to deepen understanding before problem solving (Renkl, in press-b). Objection 3:
Students usually read worked examples just superficially without striving for real
understanding. This is true (Chi, Bassok, Lewis, Reimann, & Glaser, 1989; Renkl, 1997). For this
reason, a central guideline when implementing learning from worked examples is to support
the students in explaining the rationale of the worked solutions to themselves (e.g., Renkl, in
press-b).
Mathematics and science examples with algorithmic solution procedures are only some
of the many possible varieties of solution steps. For example, Hilbert, Renkl, Kessler, and Reiss
(2008) provided the heuristic steps that students should apply when trying to find a
mathematical proof (note that such worked examples may comprise several pages); Rummel,
Spada, and Hauser (2009) have employed worked examples in the form of videos displaying
different steps in productive cooperation; Nievelstein, van Gog, van Dijck, and Boshuizen (2013)
successfully employed worked examples detailing steps to analyze legal cases. As Figure 2
shows, not all worked examples provide solution steps. Hilbert, Renkl, Schworm, Kessler, and
Reiss (2008) taught (future) teachers how to design worked examples for their mathematics
lessons by providing a well-designed and a sub-optimal worked example. The variety of possible
applications of worked examples shows that this learning method is not only feasible when
students need to learn algorithmic solutions, but also when heuristic methods (that do not
necessarily lead to an optimal solution) should be learned (e.g., Renkl, in press-b).
Another important characteristic of worked examples, illustrated by the Figures 1 and 2,
is that they often include special features to deepen students' understanding. As you can see in
Figure 1, the solution is provided in two ways and types of representations: tree diagrams and
equations. If the students are encouraged to use the tree diagrams as a “bridge” in order to see
how the problem formulations translate into the equations, their understanding and transfer
performance is greatly enhanced (Schwonke, Berthold, & Renkl, 2009). You might have also
noted that color coding (corresponding elements are displayed in the same color) makes it
easier for students to figure out which elements in the tree diagrams correspond to which
elements in the equations and vice versa. In addition, the self-explanation prompts (right side in
Figure 1) encourage interrelating the different solution procedures.
How effective are Worked Examples?
Hattie (2009) reported in his synthesis of research on learning and teaching that
conventional worked examples, when compared to learning by problem solving, have medium
to strong effects (see also the classic studies by Sweller & Cooper, 1985). Note that
conventional worked examples do not include prompts or other means to foster students' self-
explanation. When self-explanations are fostered the positive effect roughly doubles to a strong
effect (Hattie, 2009; Renkl, in press-a). If other favorable features – some of them have already
been discussed in reference to the Figures 1 and 2 – are added, the effectiveness can be further
advanced (a variety of other favorable features are discussed as instructional principles later in
this chapter).
Learning from worked examples, if implemented well, is even superior to very well-
supported learning by problem solving (Salden, Koedinger, Renkl, Aleven, & McLaren, 2010). In
many cases, worked examples not only foster learning outcomes, including transfer
performance, but also reduce the necessary learning time (Salden et al.). Zhu and Simon (1987)
found that a three-year mathematics curriculum could be taught in two years by learning from
worked examples without any performance losses. Against the background of these convincing
results, Pashler et al. (2007), in their evidence-based practice guide, count worked examples as
one of seven main recommendations for teaching and learning.
The positive effects of worked examples can be fully canceled out if important design
principles are neglected. For example, if you used worked examples such as in Figure 1 but
without color-coding and without self-explanation prompts and support, learning would very
likely not be effective; the learners can be overwhelmed with how to use and integrate the
different representations (Berthold & Renkl, 2009; Tarmizi & Sweller, 1988). In a nutshell,
learning from worked examples is highly effective but only when implemented properly.
Instructional principles that guide the implementation of this learning method will be
extensively discussed in a later section.
Why are Worked Examples Effective?
As already argued, students typically lack a profound understanding of the domain
principles and their application in the beginning of skill cognitive acquisition (Renkl, in press-b).
Hence, they are not able to rely on domain strategies for problem solving that are based on the
principles to-be-learned (VanLehn et al., 2005). Their "back-ups" are shallow strategies such as
a key word strategy (i.e., selecting a procedure by a key word in the cover story of a problem), a
copy-and-adapt strategy (i.e., copying the solution from a presumably similar problem and
adapting the numbers), or a means-ends analysis focusing on superficial problem features
(Renkl, in press-b). Not only do these shallow strategies fail to deepen domain understanding,
they bind attentional capacities for activities that are extraneous to gaining understanding (i.e.,
extraneous cognitive load). Worked examples free learners from such extraneous activities so
that there are cognitive resources for self-explanation available. By self-explaining, the students
become aware of a solution’s rationale, ideally in reference to the underlying domain
principles. After the learners have understood the domain principles and their application, they
should be encouraged to solve problems requiring the application of these principles. When
taking the reasons for the effectiveness of worked examples into account, it is becomes clear
that this learning method is only effective for the initial acquisition of cognitive skills (for a more
thorough discussion see Renkl, in press-b).
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