Using an Explicit Teaching Approach to Develop Strategic
Spirit – The Case of the Working Backwards Strategy
Yelena Portnov-Neeman & Miriam Amit
Department of Science and Technology Education, Ben-Gurion University of
the Negev, Beer-Sheva, Israel
[email protected] & [email protected]
Abstract
Learning strategies for problem solving is an important process for developing
strategic spirit. Learners who control many strategies can promote their
thinking and learning. In the current research, we investigated the use of the
working backwards strategy with talented 6th-grade math students. An explicit
teaching approach was chosen for the research. Our findings showed that
young students are capable of using the working backwards strategy in
problem solving. We also found that teaching explicitly does not fix students’
ways of thinking; on the contrary, it helps them develop their strategic spirit.
This, in turn, helps them to become active learners and fosters their
independent thinking.
Introduction
Recent studies showed that learning strategies can develop students' thinking
(Lee, 2014; Steiner, 2007). Students who behave strategically are able to
direct their own learning and acquire knowledge of a specific domain. Using
strategies often in problem solving will help in understanding how and why the
strategy works, and why it is the most appropriate way of solving a problem
(English, 1993). Students who control many strategies will become effective,
faster and more intelligent solvers (Polya, 1957). Talented students have
greater abilities to solve problems and use strategies correctly (Lee, 2014).
Tishmen, Perkins & Jay (1996) used the term "Strategic Spirit" in their studies.
They claim that students having strategic spirit can recognize a challenging
problem, prepare an action plan and execute it properly. They mention four
main reasons why the development of strategic spirit is important:
1. Helping students cope with mental barriers.
2. Strengthening independent learning: the student becomes an active
learner who examines different solutions for a problem, enabling him to
delve into the study material.
3. Fostering independent thinking: strategic spirit motivates the student to
develop his own working plan and execute it properly.
4. Providing assistance in daily situations: this is the main reason why
strategic spirit is so important. Strategic spirit helps a person make
intelligent and judicious decisions when facing different challenges. It helps
the person organize resources, prepare a working plan and carry it out
wisely.
The researchers mentioned that most students and adults lack strategic spirit
without receiving proper instruction, guidance and encouragement. This
statement motivated us to conduct the current study. We developed a learning
program in which we introduced different strategies to students for solving
math problems; one of them was the working backwards strategy. We used
different teaching approaches for the teaching strategies. Our findings from a
previous study showed that explicit teaching was a suitable and appropriate
approach for teaching this topic.
Working Backwards Strategy
The working backwards strategy is a useful and efficient strategy in many
aspects of our lives (Posamentier & Krulik, 1998; Newell & Simons, 1972).
Sometimes, the achievable outcome is known, but we have not yet determined
the path to achieve it. When dealing with word problems, the information given
in a problem can appear like a complex list of facts. In problems such as
these, it is sometimes helpful to begin with the last detail given (Shapiro, 2000;
Wrigh, 2010).
To apply this strategy, the following steps must be followed (see Fig. 1).
1) Read the problem from beginning to end and identify how many steps are
involved in the problem.
2) Check the final outcome of the problem.
3) From the final outcome, start reversing each mathematical operation in
each step until reaching the beginning of the problem.
4) Resolve the initial state.
5) Check the answer by starting from the initial state and working through the
steps to see if the final outcome is achieved (Amit, Heifets & Samovol,
2007).
Figure 1: Model of the working backwards strategy
(Amit, Heifets & Samovol, 2007).
Explicit Teaching Method
Explicit teaching is a structured, systematic and effective teaching
methodology for raising students' achievements (Archer & Hughes, 2011;
Edwards-Groves, 2002). It is called explicit because it contains a direct
approach that includes the development of guidance and an explanation of
processes, and is used mainly in the areas of reading and mathematics (Ellis,
2005; Rosenshine, 1986). A mediation process exists between the teacher
and the learner during all stages of the learning. Tetzlaff (2009) presented five
steps for teaching explicitly:
1) Orientation.
2) Presentation.
3) Structured practice.
4) Guided practice.
5) Independent practice.
The teacher is responsible for transmitting an external understanding of
information to the learner, who is then responsible for processing this predetermined understanding (Olson, 2003).
Methodology
Research Subjects
The study was carried out in the "Kidumatica" project – a mathematical club for
excellence and creativity. Kidumatica is targeted at talented 5th-11th grade
students who are interested in mathematics but require further tools, skills and
support in order to reach their full potential intellectually and cognitively,
especially in terms of their logic and mathematical skills. The project also
serves as a research model and laboratory for testing new programs and
teaching methods among gifted and talented students (Amit, 2009).
The study was carried out in two groups comprised of 50 6th-grade students for
six months. During this period, the students studied different mathematical
strategies, including the working backwards strategy. None of them had
served as research subjects in previous studies involving the working
backward strategy and they had not learned it before.
The Learning Program
In the current study, we developed a learning program comprised of different
problem solving strategies in mathematics, one of which was the working
backwards strategy. We introduced the strategy to students using the
illustrated model in Figure 1 and explained how the strategy works in different
mathematical problems. For example, the Card Problem:
Yael Danny and Michael played cards. At the beginning of the game, each
had a different amount of cards. Yael gave Danny 12 cards. Danny gave
Michael 10 cards and Michael gave Yael 4 cards. At the end of the game,
each had 20 cards. How many cards did Yael, Danny and Michael have in
the beginning?
In order to solve this problem, the students had to realize that the amount of
cards at the end of the game was known, and only by reversing each step
could they determine the amount at the beginning. The solution to the problem
involved three steps (Table 1). The students needed to realize that the amount
of cards at the end was their first clue. They had to convert each step by
making the opposite mathematical calculation.
The teacher drew a table on the board and wrote down the names of the
players. Then she wrote down the phases of the problem in their order (rows
2-4 in Table 1):
Phase 1: Yael gave Danny 12 cards.
Phase 2: Danny gave Michael 10 cards.
Phase 3: Michael gave Yael 4 cards.
The teacher wrote down the final cards amounts (row 5 in Table 1).
Afterwards, she started to solve the problem backwards, starting from the last
phase (phase 3, row 4 in Table 1) to the first phase of the problem (phase 1,
row 2 in Table 1). By doing so, the mathematical calculations were reversed.
For example, in phase 3, "Michael gave 4 cards to Yael": that means that
before this phase, Michael had 4 cards more and Yael had 4 cards less.
Therefore, the calculation to determine the amount of cards before this phase
will involve adding 4 cards to the 20 cards that Michael had (20+4 = 24) and
subtracting 4 cards from Yael (20-4 =16). At the beginning of phase 3, Michael
had 24 cards and Yael had 16 cards. The teacher then calculated the amounts
in phases 2 and 1 and determined the initial number of cards held by each
player.
Stages
The
amount
in
the
beginning
Yael gave 12 cards to
Danny
Danny gave 10 cards to
Michael
Michael gave 4 cards to
Yael
The final amount
Michael
Danny
Yael
14
18
28
14
30 – 12 = 18
16 + 12 = 28
24 -10 = 14
20 + 10 = 30
16
24 + 4 = 24
20
20 - 4 = 16
20
20
20
Table 1: Solution to the Card Problem
The students had to solve similar problems when each lesson presented more
complex problems. At first, they solved the problems with structured practice,
afterwards with guided practice, and eventually solved the problems
independently.
In the final lesson of the learning program, the teacher erased the board and
wrote a working backwards problem. This problem had the same idea as the
cards problem but with a change in numbers. The students had to solve the
problem independently.
30%
25%
20%
15%
10%
5%
0%
Solved like the Calculated the Used the table Wrote only the Wrote the
Used equation Didn't solve the
teacher
amount for
backwardly subtraction and calculations in
problem
each player
multiplication word without
separatley
mathematical
equation
Figure 2: Different methods for using the working backwards strategy
Findings from the Students’ Answers
The findings were collected from the students’ answers to the problem. We
discovered that the students used the working backwards strategy in different
ways (Fig. 2) than that presented by the teacher in Table 1.
Twenty-four percent of the students solved the problem as the teacher
demonstrated. Sixteen percent of the students did not use a table and
calculated the amount of cards separately for each player (Fig. 3). Twenty-six
percent used a table but wrote the last phase of the problem as the first phase
in the table and immediately solved it backwardly. Ten percent used a table
but wrote only the subtraction and multiplication steps without calculating the
amount in each step of the problem (Fig. 4). Six percent wrote the calculation
in words without using any mathematical equation. Six percent tried to write an
equation and solve the problem from beginning to end but struggled with it
(Fig. 5). Twelve percent were unable to solve the problem at all.
Roei: 20+8-7 =21
Anat: 20-8+13= 25
Sivan: 20-13 +7 = 14
Figure 3: Calculating the amounts separately for each player.
Sivan Anat Roei
20
20 20
-8 +8
-13
+13
+7
-7
14 25
21
Figure 4: Writing the subtraction and multiplication steps for each player.
A
b
c
Roei Anat
Sivan
a-8
b+8
c+13
a-8+7 b+8-13 c+13-7
20
Figure 5: Using questions to solve the problem.
Discussion
In the current research, we investigated how effectively young students can
implement mathematical strategies in problem solving. A learning program
was developed to this end. After several weeks of teaching the strategy
explicitly to 50 6th-graders, we examined their ability in solving working
backwards problems independently. Lee (2014) stated that talented students
have a higher ability to control strategies and use them correctly in problem
solving. Our findings showed that young talented students are more than
capable of using the working backwards strategy. They recognized the
strategy and knew when and how to use it correctly. We found out that 64% of
the students used the working backwards strategy in different ways than that
demonstrated by the teacher. They understood how to use the strategy and
develop their own plan to solve the problem, as illustrated in Figure 2. This
finding indicates that with proper instruction, students can develop their
strategic spirit (Tishmen, Perkins & Jay, 1996). This finding also indicates that
learning explicitly doesn't necessarily fix student's way of solving a problem
and thinking. Students became active learners and fostered their independent
thinking. They understood the principle of the working backwards strategy and
applied it in a way that they deemed fit.
This study could shed light for researchers and educators that strategies can
be taught even at a young age. Although mathematical strategies are not easy
to teach, the explicit teaching approach proved to be a suitable framework for
this matter. It is important that strategies be an integral part of the school
curriculum and be introduced to students already at a young age. Doing so
can promote learners’ strategic spirit and thinking.
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