THE EFFECTIVENESS AND LIMITATION OF READING AND
COLORING STRATEGY IN LEARNING GEOMETRY PROOF
Ying-Hao Cheng and Fou-Lai Lin
China University of Technology / National Taiwan Normal University
The reading and coloring (RC) strategy has been verified that it can enhanced
incomplete provers’ performance to Taiwan junior high students while they had learnt
the multi-steps formal geometry proof. In this study, we transfer the RC strategy into
whole class regular teaching and explore its effectiveness and limitation. The results
show that the learning effectiveness of RC strategy is better significantly than
traditional labelling strategy, it enhance the proof quality distribution of multi-steps
geometry proof, and it is less-effective to non-hypothetical bridging students .
INTRODUCTION
The learning and teaching of multi-steps geometry proof in Taiwan
The learning content concerning geometric shapes and solids in Taiwan is
considerably abundant in the elementary and junior high school. The geometry lessons
mainly focuses on finding the invariant properties of kinds of geometric figures and
apply these properties to solve or prove problems. The formal deductive approach of
argumentation is introduced and become the only acceptable way in the second
semester of grade 8, after introducing the congruent conditions of two triangles.
Although the manipulative approach is allowed in finding geometry properties, the
way of verifying a geometry property is basically deductive. The task of geometry
proof in formal lessons can be divided into two phases. In the beginning, the students
learn how to apply one property to show a geometry proposition is correct, that is, to
infer the wanted conclusion by one acceptable property under the given condition. We
name this is a single-step proof. In this phase, if two or more properties are necessary in
a proof question, the textbook then divide the whole question into step-by step of single
proof task. The second phase start in the final chapter of geometry lessons in the first
semester of grade 9. The students learn how to construct a formal deductive proof with
2 or more geometry properties. From chaining the step-by step of single proof into a
sequence of proof to proving an open-ended question which 2 or more properties are
necessary. That is the so-called multi-steps geometry proof question. It spend about
five weeks of regular lessons.
The teaching style in Taiwan junior high school is basically lecturing. Most of the
teachers teach geometry lessons by exposition to about 30 students in one classroom.
And the geometry proof task is basically treated as writing the reason of a proposition
by applying learnt properties.
2007. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of
the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 113-120. Seoul: PME.
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The Reading and coloring strategy
The reading and coloring (RC) strategy is initially developed for enhancing incomplete
provers’ performance in geometry proof. The incomplete provers are grade 9 students
those who had learnt the chapter of formal multi-steps proof and was able to recognize
some crucial elements to prove but missed some deductive process in 2-steps proof test.
The RC strategy ask students to read the question, label out the meaningful terms, and
then drawing or constructing given conditions and intermediary conclusions on the
given figure by coloured pens, where the congruent configurations in same colour. The
RC strategy is modified from a typical teaching strategy used in Taiwan junior high
geometry lessons which named ‘labelling’. The teachers usually use the short segments,
arcs or signs to label out the equal sides or angles between subfigures. The RC strategy
was developed based on two principles: one is it should provide an operative tool to
students for highlight necessary information, and the other one is it should keep
teachers’ regular teaching style (Cheng, Y. H. and Lin, F. L., 2006).
The function of colour and visual tool for mathematical reasoning is supported by
many literatures. Such that Byrne(1847) used coloured diagrams and symbols instead
of letters to present the formal geometry proof in ‘Elements’. He proposed that the
coloured diagrams are easier to understand the formal deductive process of Euclidean
proof. The function of this kind of visual aids was mentioned in Mousavi, Low &
Sweller(1995) they showed that a suitable visual presentation may integrate all the
information necessary in problem-solving task, reduce the cognitive load and increase
memory capacity. Stylianou & Silver(2004) find out that the difference between
experts and novices when solving advanced mathematical problem is the use of visual
representation. The experts always construct an elaborate diagram to include all the
literal information and thinking on this diagram.
From our previous studies (Cheng, Y. H. and Lin, F. L.,2005, 2006), the RC strategy is
an effective strategy to incomplete provers. Cheng and Lin(2005) showed that the
colouring the known information was effective in a highly interactive instruction. This
study showed that the intervention of colouring enhance 12/20 of not-acceptable proof
in three different unfamiliar 2-steps items to be acceptable. Furthermore, the RC
strategy enhance 14/14 of not-acceptable proof items to be acceptable in the
non-visual-disturbed multi-steps questions after about 10 minutes of teacher’s
demonstrating and 23/24 of the items in the delay post test are acceptable (Cheng and
Lin, 2006).
The aim of the study
The effectiveness of the RC strategy was verified in our previous studies which focus
on incomplete provers. They are all students who had learnt formal multi-steps
geometry proof. And these teaching experiments are conducted after the school lesson.
We cannot conclude that the RC strategy is an effective learning and teaching strategy
to all students in regular teaching. The main purpose of this study is to explore the
effectiveness and limitation of the RC strategy in regular Taiwan junior high teaching.
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THE PROCESS OF CONSTRUCTING A MULTI-STEPS PROOF
A standard geometry proof question in junior high geometry lessons and tests is of the
form ‘Given X, show that Y’ with a figure which the figural meaning of X and Y are
embedded in (fig(X,Y). When a student face to a proof question, the information
include X, Y, fig(X,Y), and the status (Duval, 2002) of X (as the premise) and Y (as the
conclusion). The proof process is to construct a sequence of argumentation from X to
Y with supportive reasons. This process can be seen as a transformation process from
initial information to new information with reasoning operators such as induction,
deduction, visual judgment… (Tabachneck & Simon, 1996). The acceptable reason in
the junior high geometry proof lessons is deduction with acceptable properties. So, we
may say that to prove is to bridge the given condition to wanted conclusion by
acceptable mathematical properties.
Healy & Hoyles(1998) propose that the process of constructing a valid proof involves
several central mental processes:(1)students might sort out what is given properties
already known or be assumed and what is to be deduced;(2)students might organize the
transformation necessary to infer the second set of properties from the first into
coherent and complete sequence. Duval(2002) propose a two level cognitive features
of constructing proof in a multi-steps question. The first level is to process one step of
deduction according to the status of premise, conclusion, and theorems to be used. The
second level is to change intermediary conclusion into premise successively for the
next step of deduction and to organize these deductive steps into a proof.
In a single step proof question, the process is relatively simple. The student might
retrieve a property ‘IF P then Q’ which condition P contain the premise X and result Q
contained in Y and finish the proof. We may say this kind of bridging is simple
bridging.
The proof process in a multi-steps proof question is much more complex. Since there is
no one property can be applied to bridge X and Y. The student has to construct an
intermediary condition (IC) firstly for the next reasoning. The IC might be reasoned a
step forwardly from X. It is an intermediary conclusion (Duval, 2002) inferred from X
as a new premise to bridge Y. Or, it might be reasoned a step backwardly from Y. It is
a intermediary premise reasoned from Y as the wanted conclusion to bridge X. So, the
first step in a multi-step proof is quite different to the step in a single step proof. The
first step in a multi-step proof may be a goalless inferring from X and concluding many
reasonable intermediary conclusions. The next step is to go on the bridging process to
Y by selecting a new premise from the intermediary conclusions. Or, it may be a
backward reasoning from Y and finding many reasonable intermediary premises and
the next step is to set up a new conclusion from the intermediary premises and going on
the bridging process from X. In fact, this kind of reasoning is lasting before the final
step of completing a proof. No matter this kind of reasoning is constructed by forward
or backward reasoning, constructing the intermediary condition in a multi-steps proof
is essentially a process of conjecturing and selecting/testing. We may say this kind of
reasoning process is hypothetical bridging.
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In summary, constructing an acceptable geometry proof can be seen as a bridging
process from given condition to wanted conclusion with inferring rules. The necessary
process includes (1) to understand the given information and the status of these
information, (2) to recognize the crucial elements which associate to the necessary
properties for deduction, (3) especially in multi-steps proof, to construct intermediary
condition for the next step of deduction by hypothetical bridging, and (4) to coordinate
the whole process and organize the discourse into an acceptable sequence.
STUDY DESIGN
Transferring the RC strategy into regular teaching
Our previous studies applied RC strategy to enhance incomplete provers’ performance
in geometry proof. This type of students have learnt the formal multi-steps geometry
proof and are able to recognize some crucial elements and construct meaningful
intermediary condition. And in our experiments, these students are grouped into
individual interview or small group learning. The above conditions are quite different
to real classroom teaching.
We design the whole class teaching with RC strategy based on two principles. The first
is it should be easily to apply by the teacher. Since it is not easy for them to apply a
completely new method in regular teaching, the strategy should be easy to fit into
teachers’ typical teaching approaches (lecturing). The RC strategy is modified from a
typical teaching strategy used in Taiwan junior high geometry lessons which named
‘labelling’: using the short segments, arcs or signs to label out the equal sides or angles
between subfigures. In this study, we ask the teacher use different coloured chalk pen
to show the process of proving in the RC strategy on the blackboard: drawing or
constructing given conditions and intermediary conclusions on the given figure by
coloured chalk pens, where the congruent configurations in same colour.
The second principle is it has to be practicable use to students in all proving task,
including taking notes, exercises, homework. In this study, we provide every student a
8-color pen and ask them to use it in all proof task mentioned above.
The samples
A questionnaire with four items are developed and tested as pre-test in 4 classes of
grade 9 students before they learn the chapter of formal multi-steps geometry proof.
Two of the items are single step and two are multi-steps. The students’ performance in
these items are coded into acceptable, incomplete, improper, intuitive response, and no
response five types according to the coding framework developed in the national
survey (Lin, Cheng and linfl team, 2003). The average score of these four classes in the
school tests of geometry lessons are considered. Two of these four classes are selected,
one for the RC strategy and the other for the traditional labelling strategy, for this study
because their performance in the pretest and score of school tests are not different
significantly.
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The mathematics teachers of these two classes are both experienced teachers. We
divide them into two different teaching groups because the teacher of RC class (T1)
accepts our suggestion of applying the RC strategy into her regular teaching and the
teacher of labelling class (T2) refuses.
The process
In the beginning, we show a demo of the RC strategy by video type and the
effectiveness of the RC strategy from our previous study to both teachers. We discuss
the function and possible procedure for using this strategy in the five weeks, whole
class teaching. Since the teacher T2 refuses to apply the RC strategy in his teaching, we
then go on our study with only teacher T1.
During the 5 weeks of teaching the chapter of formal multi-steps geometry proof, T1
uses different coloured chalk pen to show the process of proving: she draws or
constructs given conditions and intermediary conclusions on the given figure by
coloured chalk pens and uses the same colour in the congruent configurations. At the
same time, she asks her students imitate her way in all proving task: taking notes, doing
exercises and homework. She checks students’ work carefully and ask her students
strictly to use this strategy. We provide every student in RC class a 8-color pen. The
classroom activities are video typed and students’ manuscripts are photographed.
Just after the school test of the chapter of formal multi-steps geometry proof, a post test
is conducted for both the RC class and the Labelling class. We compare the leaning
effectiveness of the RC strategy in both quality type of proof from the post test and the
score from the formal school test. The items of post test are composed with four
multi-steps geometry proof questions. These items are used in our previous studies
(Cheng, Y. H. and Lin, F. L., 2006). Fig1 is the item used both in the pretest (item 4)
and post test (item 1). We use this item to explore what happens from the beginning to
the end of the teaching.
Fig1. The 2-steps item for comparison
RESULTS AND DISCUSSION
The learning effectiveness of RC is better significantly than traditional Labelling
The score of school test after the teaching and the performance in the post test are
shown in Table 1. Table 1 show that the score of school test after the multi-steps
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lessons and the quality distribution of the post test of RC class is significantly better
than Labelling class. The score of school test is significantly better and the percentage
of acceptable type in all items is too. This result shows that the RC strategy can be
applied in whole class regular teaching. And its effectiveness is significantly better
than traditional labelling strategy in both proof quality and formal school test.
class
RC class
Labelling class
49.21
43.73
School tests Standard deviation 24.54
19.50
Score of
Average
item
1
2
performance No response
0.0
39.4 12.1 6.1
in the
Intuitive response
12.1 0.0
Post test
Improper
15.2 15.2 39.4 36.4 9.1 33.3 60.6 39.4
(percentage) Incomplete
Acceptable
12.1 6.1
3
0.0
6.1
4
0.0
1
2
3
4
18.2 21.2 9.1 9.1
12.1 3.0 6.1 0.0
15.2 30.3 24.2 15.2 24.2
60.6 39.4 42.4 42.4 30.3 18.2 9.1 27.3
Table 1: The performance of the samples in the pretest
The RC strategy enhance the quality distribution of multi-steps geometry proof
According to Lin, F. L.; Cheng, Y. H. & linfl team(2003), there is about one quarter of
Taiwan junior high students, while they finish the formal multi-steps geometry proof
lessons, can construct acceptable proof in an unfamiliar 2-steps question. More than
one-third of them do not have any response. And approximately one third of them are
incomplete. We use this item of the national survey both in pretest (item 4) and post
test (item 1). Comparing the distribution of quality type from this study and the result
of the national survey (shown in Table 2), we can find out that the percentage of
incomplete type in the RC class is significantly less than in the national survey. And
there is no more performance of type of no response. This result show that the RC
strategy may help many of the ‘potential’ incomplete provers to overcome their
learning difficulties when learning multi-steps geometry proof in the traditional
teaching.
Furthermore, there is no more performance of type of no response is meaningful. It
shows that the RC strategy may help some ‘potential’ no response students to
recognize some meaningful information and start to prove. Many students can not start
to prove because they can not find out any information associate to a suitable
mathematical property. The coloured subfigure may provide more information which
is implicit in traditional teaching.
In conclusion, the RC strategy shows the subfigure which associate to a geometry
theorem and keeps all information visible and operative. It is helpful to retrieve
suitable theorem for reasoning and also helpful to reduce the memory loading when
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organizing several steps into a proof sequence. Duval(2002) proposed that retrieving
the suitable theorem is one of the key processes in geometry proof and this process is
highly depend on the theorem mapping. This study shows that the theorem is easier to
retrieve when the correspondent subfigure is highlight by colouring.
acceptable Incomplete Improper
National sample 24.6
RC class
in the post test
60.6
Intuitive
No
response
response
35.0
0.3
2.8
37.4
12.1
15.2
12.1
0.0
Table 2: The distribution of proof quality in the national survey and RC class
RC is less-effective to non-hypothetical bridging students
Even the results in this study show that the RC strategy is more effective than
traditional labelling strategy in whole class regular teaching, there are nearly 40% of
students can not construct an acceptable proof. In order to investigate the limitation of
the RC strategy, we conduct a post analysis. We re-code the manuscripts of the pretest
by considering the performance of hypothetical bridging. The hypothetical bridging is
a necessary process when proving a multi-steps geometry proof. This process construct
the intermediary condition (IC) and motivate the second level (Duval, 2002) of proving.
We categorize students’ performance of the two multi-steps questions in the pretest in
to three type: (1)hypothetical bridging, it means that the students construct some
intermediary conditions for the next step of reasoning, no matter the IC is useful or not
in that question; (2)simple bridging, it means that the students finish the proof by
applying only one mathematical property, and it is of course not correct; (3)no
reasoning, such that no response, transcribing the item. The performance of these three
types of students in the post test is shown in Table 3. Table 3 shows that all the
acceptable proof comes from the students who are able to prove a multi-steps question
by hypothetical bridging. There are 19/23 of hypothetical bridging students can
construct an acceptable proof and no one of non-hypothetical bridging students can di
it. It is obviously that the RC strategy can not help the non-hypothetical bridging
students to enhance their proof quality more. Since the main function of RC strategy is
showing the subfigure which associate to a geometry theorem and keeping all
information visible and operative, it is more useful in the information processing
process than overcome the cognitive gap. It may help students to retrieve suitable
theorem for reasoning and also helpful to reduce the memory loading when organizing
several steps into a proof sequence but if the students understanding of geometry proof
is only restricted in the first level (Duval, 2002) of proving, that is applying one
theorem to bridge the premise and conclusion, then the coloured figure may only help
the student to find out the first step (and only one step to him/her) to prove.
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Intuitive
response
No reasoning
1
Simple bridging
3
Hypothetical bridging
Improper
Incomplete Acceptable
1
4
1
1
3
19
Table 3: The performance of three reasoning type in the post test
References
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Cheng, Y.H. & Lin, F.L (2005) One more step toward acceptable proof in geometry.
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