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Assessing High School Studentsβ Ability in Proving Math Statements and
Reclaiming Its Importance
Iwan PRANOTO
pranoto@math.itb.ac.id
Department of Mathematics, Institut Teknologi Bandung, Bandung, Indonesia
Abstract
In Indonesia, standardized test is utilized not only for the evaluation of the
learning and teaching process, but it is also utilized for deciding if a student
can graduate or not. Every student in grades 6, 9, and 12 is required to pass
the exam. This makes the test becomes a high-stakes test. Therefore, the
students learn for the test, and the teachers teach for the test. It is reasonable
to think that this practice will make students ignore the skills untested in the
exam. In 2012-2013, fourteen studies were conducted to find out if proofs are
erased in the school math instruction and, in particular, to assess the
studentsβ ability in proving math statements. It was found that most students
and teachers neglected the proofs. In this paper, some diagnostic results
regarding the studentsβ ability or inability in proving are presented. Some
ideas to teach proof and its assessment are proposed.
Keywords: proof; assessment; reasoning
Introduction
Most mathematicians agree that proof is the very foundation of mathematics. Doing
mathematics must include some proving. Perhaps it is not limited to the proof of a theorem or
lemma, but some kind of formulation of arguments and a line of reasoning are vital. Students
must understand and be convinced that proving mathematics is different from other subjects
like physics. In mathematics, students as early as elementary level should realize that to say a
mathematical statement is βtrueβ, or more precisely valid, they must provide supporting
reasoning, based on deductive reasoning. Even though, mathematicians use inductive
reasoning in their work, but they only rely on deductive reasoning when determining if a
statement is valid, or not. Without proof, mathematics is not complete.
In NCTMβs Process Standards [NCTM, 2013], in Reasoning and Proof domain, it is clearly
stated that students must enable students to
β’
β’
β’
β’
Recognize reasoning and proof as fundamental aspects of mathematics
Make and investigate mathematical conjectures
Develop and evaluate mathematical arguments and proofs
Select and use various types of reasoning and methods of proof
Reasoning and proof are fundamental parts of school mathematics. However, in reality, proof
is not regarded as important. According to Knuth (2002b), proof has become βperipheral at
best, usually limited to the domain of Euclidean Geometry.β Thus, proof has not only been
considered unimportant, but it has also been neglected in most areas of school mathematics
outside of Euclidean Geometry nowadays. In Indonesia, the situation is not so different.
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Based on a number of studies done in 2012-2013, such as [Ali, 2013], [Gatot, 2013], [Inaku,
2013], mathematical proofs have been considered less important.
The studies include questions on why the teachers neglected the proving in the classroom.
One common reason they all pointed at is the national standardized exam. In Bahasa
Indonesia, it is called Ujian Nasional. It should be noted that the actual primary purpose of
this standardized exam is to determine the pass-fail of every student. Every student in their
final year of their elementary and secondary, grades 6, 9, and 12, is required to pass the exam.
Even though the score is weighted to 60%, not absolute, this exam is still a high-stakes exam.
Students, teachers, schools, and even parents prepare carefully for this exam. The claim that
this exam is a diagnostic type is not convincing, because until now the Research and
Development body has never released the analytical results.
Until now the exam has never posed problems requiring students to prove. This is the very
reason, the teachers interviewed said that they did not teach mathematical proof because it
was never assessed in the exam. The teachers asked why they should teach reasoning and
proof if the national exam has never assessed these things. This strengthens what Steen
[Steen, 1999] says that students and the educators value what the assessment assesses. Since
we math educators are not bold enough to assess what we value, like mathematical proof, the
students will value merely what the assessment assesses. Thus, we know why proof becomes
unimportant in Indonesia school mathematics practices.
However, to assess proving ability is not an easy thing to do. In particular, because
standardized test is becoming so popular and mass assessment becomes the norm, proof and
reasoning in school mathematics are not in the top list of skills. This paper try to propose a
kind of math test that assesses studentsβ ability in reasoning and proof. At the same time, the
test should be easy enough to be administered for large class size. This proposed test is also
an evidence stating that assessing the proof and reasoning is possible in the standardized
exam. In addition, there was a popular belief among math teachers that assessing the proof
and reasoning must use essay type of tests.
Proof in Mathematics
In general, it is accepted that mathematical proof is considered an important part in school
mathematics. At least formally, proof is believed and considered one of the basic skills that
must be developed. For example, in the Content Standards of the Indonesian Body of
National Educational Standardization (BSNP), it is stated that through mathematics, all
students must have capability to βuse reasoning on patterns and properties, do mathematical
manipulation in order to make generalizations, compose mathematical proof, or explain
mathematical ideas and statements.β
However, unfortunately, it is found that on most basic competence lists in the Indonesian
Math Standards, the mathematical proof is not elaborated explicitly enough. The verbs used
in the basic competence list are mostly on the application of the formula. Like Pythagorean
formula, for example, it is stated only that students must be able to use the formula for
solving problems. In the Remainder theorem is also the same. It is stated that students are
expected to be able to use the theorem. It is not stated whether the students must be able to
prove or at least be familiar with the proofs.
In the studies conducted it was found that even some teachers interviewed were not familiar
with the proofs. Some of the teachers said that teaching mathematical proofs is time
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consuming. Of course, it confirms our prior beliefs that mathematical proof is not addressed
sufficient enough in school math practices.
In addition, we also found that most school math textbooks did not mention mathematical
proof. If there were, like for the triangle sum theorem, it was βprovedβ empirically. It asks
students to make a paper triangle and then ask them to cut the corners. After that, it asks the
students to glue the three pieces together to make a straight line or a 180-degree angle.
Figure 1. The βproofβ in most math books [Gatot, 2013:3]
We conducted studies on some theorems, formulas, procedures, and lemmas in secondary
mathematics. The topics studied include Pythagorean theorem, Triangle sum theorem,
Remainder theorem, Division with fractions, Derivative formula for algebraic expressions,
etc.
Diagnostic Findings
There are two findings. First, our studies confirm that the students were lack of the ability to
prove and reason. Second, the actual math teaching processes did not involve mathematical
proof. Moreover, the math reasoning was not promoted enough in practices.
When we study the studentsβ understanding on the proof of Pythaorean theorem, it was first
found that most Indonesian βBSNP-approvedβ math textbooks present the example as a
βproofβ. For example, a special right triangle whose sides 3, 4, 5, respectively, in Figure 2
below is used to βproveβ the theorem [Inaku, 2013:12].
There is nothing wrong with presenting special cases, but the word βproofβ in mathematics is
totally different from giving illustrations. It is also true that unfortunately one cannot prove
all statements in school mathematics using K-12 mathematics. However, math educators
must use the word βproofβ and βproveβ only when they use deductive reasoning. At the same
time, when they just provide illustrations confirming some statements, they should just say
the illustrations support the conjecture.
Very often, school math textbooks present mathematics not rigorously enough. Some books
are written not in math standard level. It is precisely like Wu says in [Kiang, 2012] that
βEverything is informal; it almost never gives a precise
definition.β
He continues to say that some school math textbooks misrepresent mathematics in the sense
that they fail to
β(1) give precise enunciations of definitions and mathematical
conclusions (i.e., theorems),
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(2) provide reasoning to support each mathematical assertion,β
So, in short, there must be a rigorous separation between proof and conjecture in school
mathematics textbooks and
Figure 2. "Another way to prove the Pythagorean theorem is by putting a square on each side of the right
triangle...β
Of course, the proofs are available and easily accessible on the Internet. Moreover, some
math textbooks such as [Larson, 2001:535] do give the proof in a great detail. It is complete
with the explanation on each step. In addition, the theorem and the proof are presented in a
rigorous form. The theorem is boxed accordingly and the language is both precise and
proper. The question is whether these resources have been accessed and utilized by the
students.
Perhaps it is related or not, but the student subjects observed did show some lack of precise
understanding on definitions and theorems. For example, when some students were asked to
identify the remainder in an expression like π(π₯) = (π₯ 2 + 1)(π₯ + 1) + (π₯ 3 + 1), the
subjects instantly pointed to the last term as the remainder [Ali, 2013:28]. At the same time,
the definition of the remainder was not precise enough in most Indonesian secondary school
textbooks we studied. We do not know whether these two situations have cause-and-effect
relationship.
Moreover, when the subjects were asked to solve a problem in a non-routine form, they
showed some difficulties. For example in [Ali, 2013:28], the subjects were asked to
determine the remainder if (π₯ 20 β 1) is divided by (π₯ 2 β 3π₯ + 2). All subjects tried to solve
directly using several division algorithms, like in Figure 3. Not surprisingly, they were stuck
in the middle of the long division process. It seems that the subjects could not relate this
problem with the Remainder theorem. The situation showed that the subjects had not been
able to understand deeply on what division of polynomials is. The conceptual understanding
behind this theorem seemed was not well comprehended.
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Figure 3 Students's work on the Remainder theorem
Some observations like in [Murniati, 2013] state that some students misunderstood the role of
illustrations. The student subjects thought that a great number of examples are enough for
proving the validity of a mathematical statement. This misunderstanding of course is not the
sole results of secondary school mathematics teaching. It is probably the elementary school
math teaching practices in the elementary school contributes to this foundational
misunderstanding too.
The same study also informs that the student subjects tend to memorize the concepts instead
of deeply understand them. Because this surface strategy is sufficient enough for passing the
routine exams, the subjects became more and more like to memorize things. Because of this
βlearningβ strategy, students became easily forget and not able to solve non-routine problems.
Logically, this unfortunate situation will make the level of self-esteem of the students in
mathematics decrease.
Assessing Reasoning and Proof
Proof must be assessed. Math educators need to understand the progress of math learning of
their students. So, proof must be learned and teachers must teach it. However, students must
enjoy the learning process. Proving in mathematics is a very sophisticated task, it requires
very high order of thinking. So teachers must assist and empower their students in the
learning process.
Even though math teachers are not encouraged to soften down the difficulty of math proof,
they must create fruitful and enjoyment in the classroom atmosphere. This is the vital role of
math teachers nowadays. In [Quinn, 2012] it is stated that βIt is important, therefore, to
carefully design examples and procedures to guide effective learning.β
Some assessment can be used to evaluate studentsβ capability to prove. Contrary to the
popular beliefs that say mathematical proof can be assessed only in essay type tests, during
the series of studies, several type of tests we tried.
One way to assess the proving capability is to ask the students to put in order a series of
arguments. Prior to that, a complete proof is cut into several pieces of arguments. The
students must sequence the pieces together to make a solid proof of the statement. For
example, in [Gatot, 2013:50], it is shown how to assess the capability of the students to prove
the triangle sum theorem. So, this kind of assessment like in Figure of course can be adapted
for a multiple-choice type test. It can of course easily be adapted for even online type test.
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Figure 4 Assessment on math proof by ordering the arguments
The dotted lines were cut. The student subjects then were given those pieces. This kind of
assessment can be applied to all kind of math proofs.
Another type of assessment is to complete the reasoning for each step of proof. So the
students are presented with a not-so-complete proof. In particular, the steps of the proof are
complete, but the reason why each step is allowed and valid is not given. So the students
must select on the given list of reasons. They must select which argument is appropriate for
each step. The students can match each step with the correct argument.
Figure 5 Assessing math proving capability by matching arguments
One can hope that if reasoning and proof are assessed in the regular test, they will be
developed in the classroom and outside. In one of the studies conducted, students were taught
to understand math proof and write ones. After the lesson, students were asked to prove some
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mathematics statement [Ali, 2013:36], that is to prove βIf the polynomial π(π₯) is divided by
(π₯ + π), then the remainder is π(βπ).β One of the proofs given by the subject is very
readable. The translation is the following:
Proof. Let π(π₯) be the polynomial. It is divided by (π₯ + π) and let the
result be β(π₯) and the remainder be π .
So, it can be written as π(π₯) = β(π₯)(π₯ + π) + π .
Therefore, when π₯ = βπ, it becomes π(βπ) = β(βπ)(βπ + π) + π .
So, π(βπ) = 0 + π . Thus, π(βπ) = π .
Because π is the remainder, then the remainder is π(βπ).
So, if π(π₯) is divided by (π₯ + π), the remainder is π(βπ).
β
Conclusion
What the high-stakes exams assess will determine what the students and teachers value. In
particular, reasoning and proof while they are vital in mathematics, they are avoided in
classrooms nowadays, because they are not assessed in the exam. When most popular beliefs
say that reasoning and proof cannot be measured through standardized tests, the studies in
this research suggest otherwise. Moreover, while this qualitative studies are limited in size,
one can broaden the scale to test for a bigger population. Lastly, students are very capable to
do reasoning and compose mathematical proofs.
References
Ali, S. (2013). The learning of the Remainder theorem concept on polynomial division with
the divisior (ππ₯ β π). Final project report, Mathematics Master Program, ITB.
Gatot, H. (2013). Assessing the proficiency to prove the sum of interior and exterior angles of
a triangle. Final project report, Mathematics Master Program, ITB.
Inaku, R. (2013). Teaching mathematical power on some students through the proof of the
Pythagorean theorem. Final project report, Mathematics Master Program, ITB.
Kiang, L. Y. (2012). Mathematics K-12: Crisis in education. Mathematical Medley, Vol. 38,
No. 1, June 2012, pp 2 β 15.
Knuth, E. J. (2002a). Teachersβ conceptions of proof in the context of secondary school
mathematics. Journal of Mathematics Teacher Education, 5, 61-88.
Knuth, E. J. (2002b). Secondary school mathematics teachersβ conceptions of proof. Journal
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Larson, et. al. (2001). Geometry. McDougall Littell.
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NCTM
(2013).
Process
Standards.
NCTM.
Accessed
July
25,
2013.
http://www.nctm.org/standards/content.aspx?id=322
Quinn, F. (2012). Contemporary proofs for mathematics education. Proceedings of the ICMI
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Steen, L. A. (1999). Assessing assessment. In Assessment Practices in Undergraduate
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Steen, L. A. (2007). Moving beyond standards and tests. Invited testimony at the initial
meeting of the Commission on Mathematics and Science Education, established by the
Carnegie Corporation of New York and the Institute for Advanced Study in Princeton,
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Wu, H. (2009). What's sophisticated about elementary mathematics? American Educator,
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