RENEWING THE PROVING EXPERIENCES: AN
EXPERIMENT FOR ENHANCEMENT TRAPEZOID AREA
FORMULA PROOF CONSTRUCTIONS OF STUDENT
TEACHERS BY “ONE PROBLEM MUTIPLE SOLUTIONS”
Xuhua Sun
Education faculty, University of Macao, China
Abstract
This study is to explore the possible opportunities for student teachers to
acquire the experience necessary to provide effective instruction about proof
and proving. The 14 student teachers from the University of Macao were
required to prove “area formula of trapezoid” with “one problem multiple
solutions”. The results revealed that their own original proof constructions
were generally enhanced. Twelve creative methods of proving this formula
were generated. The notion of one problem multiple solutions should be
uplifted into a curriculum design framework guiding our teaching practice
was discussed.
Introduction
Proof undoubtedly lies at the heart of mathematics. Proof has played a major
role in mathematics. The learning of proof and proving in school mathematics
would clearly depend on teachers’ views about the essence of proofs, on what
teachers do with their students in classrooms that have the potential to offer
students opportunities to engage in their own original proof constructions.
However, Student teachers as pre-service teachers had learned most of
fundamental mathematics formulas and theorems they will teach latter. When
they teach these formulas and theorems, they used to unconsciously recollect
old proofs copied from their textbooks or past-experiences. The ready-made
solutions of formulas and theorems proof have impeded their exploration of
own original proof constructions, which would shape their proof teaching
practices into superficial and imitate proving in place of opportunity for
development their students` original proof constructions. In other hand, there
is a pressure on textbooks to be self-contained (so students do not have to ask
the teacher many questions) by providing guiding questions, which have also
impeded their exploration of own original proof (Lithner, 2003). However, to
teach original proof, one should first know what original proof is. The
teachers’ deficiency in understanding how to construct own original proof
determined their inability to teach original proof construction and would have
no real practice in teaching original proving . Even their pedagogical
knowledge could not make up for their ignorance experiences of own original
proof.
“Area formula of trapezoid” is a basic formula to calculate the area of
trapezoid. Most of curriculum materials heavily focus on memorizing by rote
and mechanically applying the formula, rather than own original proof in
most counties. It should eventually have impeded the building proper
conceptions of math learning in the long run.
It is impressive to note that the method in the USA textbook(Bolster, Boyer,
Butts, & Cavanagh , 1996, page 350) presented one justifying methods alone
by illustration (The two same trapezoids are reorganized into a parallelogram)
However, Chinese textbook (Mathematics textbook developer group for
elementary schools, 2003, p.88) presented three justifying methods by
illustration (The trapezoid is separated into two triangles; the trapezoid is
separated into a triangles and a parallelogram; the two same trapezoids are
reorganized into a parallelogram.) It seems that Chinese textbook was better
at using “one problem with multiple solutions” than the US counterpart in this
case. The prior study (Sun, 2007) found that “one problem multiple solutions”
is widespread, and well known, in China but still far from uplift into a
curriculum design framework guiding our teaching practice. “One problem
multiple solutions” could be regarded as an effective tool to guide students to
explore own methods. We wonder whether student teachers may improve
own proving of trapezoid formula proof constructions by “one problem
multiple solutions”. In this study, the student teachers were asked to prove
“area formula of trapezoid” with one problem multiple solutions. The aim of
design tended to change their habit to recollect old proof copied from their
curriculum or past-experiences by regenerating their own proving
experiences.
Methodology
About 14 Students from University of Macao were asked to prove “area
formula of trapezoid” with their own methods for 2 hours and then wrote
down their own methods of proving on the blackboard one by one. Each a
method was named after their first names. Their drafts and the whole process
videotaped were collected for further analysis. All methods in the blackboard
were taken photos by a camera.
RESULTS
The results revealed the student teachers had ability for their own original
theorem proof constructions. The 12 creative methods 1below of proving
this formula were regenerated.
1. Method of Can
Connect AC. The triangle ∆ABC and ∆ACD
same height h，
So
have the
S ABCD = S ∆ABC + S ∆ACD
=
ah bh ( a + b) h
+
=
2
2
2
COMMENT：This is a simplest proving method among
all methods presented by the textbooks of different
countries.
2. Method of Bin
E is the mid point of CD. Connect AE and BE. So,
S ABCD = S ∆ADE + S ∆ABE + S ∆BCE
=
1 b
ah 1 b
( a + b)h
⋅ ⋅h+
+ ⋅ ⋅h =
2 2
2 2 2
2
COMMENT：The trapezoid is divided into 3 triangles.
The key point of the method is finding of midpoint,
which make proving simple .Of course, any a point on
the line DC is an available too.
3. Method of Zhu
E is midpoint of BC. Connect AE. F is the
intersection of extended line DC and extended line
AE.
∠ABE = ∠FCE
 AB = FC

⇒ ∆ABE ≅ ∆FCE ⇒ 
 BE = CE
S ∆ABE = S ∆FCE
∠BEA = ∠CEF

Then
1
I present 7 methods here due to no room for them.
S ABCD = S ∆ADF =
( a + b) h
2
COMMENT：The trapezoid is skillfully transformed
into a triangle with same area by replacing ∆ABE
by∆FCE. It is a creative proving.
4. Method of Chan
Extend BA and DC. E is intersection of BA and CD.
Draw height EG and height AF. G is the intersection
of EG and BC. F is the intersection of AF and BC.
Because AD // BC ，the triangle EAD is similar to the
triangle EBC，
EH
EH
AD a
=
=
=
EG h + EH BC b
ah
Then EH=
b−a
S ABCD = S ∆EBC − S ∆EAD
=
a( h + EH ) bEH ( a + b) h
−
=
2
2
2
COMMENT：The trapezoid is extended into a
triangle by extending its two sides. The EH was
eliminated according to the property of the similar
triangle.
5. Method of Xian
Extend AB to E，so as to BE = CD . Extend DC to
F，so as to CF = AB . Then AE = FD
and AE // FD . So AEFD is a parallelogram.
S ABCD =
1
( a + b) h
S AEFD =
2
2
COMMENT：The trapezoid is reorganized into a
parallelogram by copying the same trapezoid.
6. Method of Feng
Draw CE // DA such that line CE passes through
point E. Then we have AECD is a parallelogram.
S ABCD = S AECD − S ∆BEC
= bh −
(b − a ) h ( a + b) h
=
2
2
COMMENT：The trapezoid is reorganized into a
parallelogram by making a parallel line.
E
D
A
H
B
F
G
C
7.Method of Yu
Draw the symmetry points A` and B′of A and B
based on symmetry axis DC
Then
1
S ABCB' A' D
2
1
( a + b) h
= (SCDGH + S A' B ' FE ) =
2
2
S ABCD =
COMMENT：The trapezoid is reorganized into 2
rectangles by making a symmetry figure.
Here are some photos about students` proving solutions presented at the
classroom. (See figure 1).
Figure 1 THE PHOTO OF STUDENTS` SOLUTIONS
IMPLICATION
Why did we stress “one problem multiple solutions”?
“one problem multiple solutions” is one of frameworks in spiral variation
curriculum, specially filtered and rationalized problem variations with
multiple conceptions connection or multiple solutions connection2 from
Chinese own teaching experience from its own mathematics curriculum
practice, was tried out in 21 classes at the primary schools in Hong Kong,
The effect of curriculum are significant (Sun, 2007; Wong, 2007).It could be
traced to prior variations study, which are identified as an important element
of learning / teaching mathematics in China by some researchers, educators,
and teachers in recent years (Gu, Huang & Marton, 2004; Sun，2007; Wong,
2007). The study indicated that “one problem multiple solutions” (called
problem variations with multiple solutions connection in spiral variation
curriculum) just successfully helped student teachers renew their proof
experience and further reconstruct their own solution system to some extent,
2 The former one is called One problem multiple variation , “ 一题多变” in Chinese, varying conditions,
conclusions. The latter variation is called as one problem multiple solutions i.e. “yiti duojie” “一题多解” in
Chinese , varying solutions.
which tends to display some advantages3 in changing the habit of superficial
and imitate proving , lead to long term gains, like improving interest,
self-efficacy and independent analysis in the central roles of mathematics
learning. The results make us realize that “one problem multiple solutions”
is not fully apprehended by students, teachers, textbook writers, and perhaps
also among many researchers. One reason may be that we lack exploration
deep its effectiveness to extend students` method system and gain the
original insights in more specific and real ways due to too familiarness.
“One problem multiple solutions” is a simple and powerful framework
for guiding teaching and learning. In fact the notion of one problem
multiple solutions is widespread, and well known, just like air we breathe we
seldom are aware of its existence. But the notion of one problem multiple
solutions is still far from uplift into a curriculum design framework guiding
our teaching practice. Just to refer that practice with one problem multiple
solutions will not help much, if we cannot specify a curriculum design frame
and further probe its effectiveness. This case show it can take us far, and
indeed it used to do.
REFERENCES
Bolster, L. C., Boyer, C., Butts, T., & Cavanagh, M. (1996). Exploring
mathematics (Grade 7). Glenview, IL: Scott, Foresman.
Gu, L., Marton, F., & Huang, R. (2004). Teaching with Variation: A Chinese
Way of Promoting Effective Mathematics Learning (pp. 309-347). In L.
Fan, N. Y. Wong, J. Cai, &. S. Li (Eds.), How Chinese learn mathematics:
Perspectives from insiders. Singapore: World Scientific.
Lithner, J. (2003). Students mathematical reasoning in university textbook
exercises. Educational Studies in Mathematics, 52, 29–55.
Mathematics textbook developer group for elementary school. (2003).
Mathematics. upper volume, Grade 4, Page 88 [in Chinese]. Beijing:
People’s Education Press.
Sun, X.H.(2007). Spiral Variation (Bianshi) Curriculum Design In
Mathematics: Theory and Practice. Unpublished doctoral dissertation,
Hong Kong: The Chinese University of Hong Kong.
Wong, N. Y. (2007). Confucian Heritage Cultural learner’s phenomenon:
From “exploring the middle zone” to “constructing a bridge”. Regular
lecture, the Fourth ICMI-East Asia Regional Conference on Mathematical
Education, Penang, Malaysia, 18th to 22nd, June.
We also did other experiments of more than ten theorems and formulae by one problem multiple
solutions. The whole effect is inspiring and significant.
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