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Assessing High School Students’ 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 students’ ability in proving math statements. It was found that most students and teachers neglected the proofs. In this paper, some diagnostic results regarding the students’ 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 “true”, 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 NCTM’s Process Standards [NCTM, 2013], in Reasoning and Proof domain, it is clearly stated that students must enable students to

• 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 “peripheral at best, usually limited to the domain of Euclidean Geometry.” 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 students’ 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 “use reasoning on patterns and properties, do mathematical manipulation in order to make generalizations, compose mathematical proof, or explain mathematical ideas and statements.” 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 “proved” 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 “proof” 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 students’ understanding on the proof of Pythaorean theorem, it was first found that most Indonesian “BSNP-approved” math textbooks present the example as a “proof”. For example, a special right triangle whose sides 3, 4, 5, respectively, in Figure 2 below is used to “prove” the theorem [Inaku, 2013:12]. There is nothing wrong with presenting special cases, but the word “proof” 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 “proof” and “prove” 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

“Everything is informal; it almost never gives a precise definition.”

He continues to say that some school math textbooks misrepresent mathematics in the sense that they fail to

“(1) give precise enunciations of definitions and mathematical conclusions (i.e., theorems),

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(2) provide reasoning to support each mathematical assertion,”

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...” 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 𝑝(𝑥) = (𝑥2 + 1)(𝑥 + 1) + (𝑥3 + 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 (𝑥20 − 1) is divided by (𝑥2 − 3𝑥 + 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 “learning” 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 “It is important, therefore, to carefully design examples and procedures to guide effective learning.”

Some assessment can be used to evaluate students’ 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 “If the polynomial 𝑝(𝑥) is divided by (𝑥 + 𝑎), then the remainder is 𝑝(−𝑎).” One of the proofs given by the subject is very readable. The translation is the following:

Proof. Let 𝑝(𝑥) be the polynomial. It is divided by (𝑥 + 𝑎) and let the result be ℎ(𝑥) and the remainder be 𝑠.

So, it can be written as 𝑝(𝑥) = ℎ(𝑥)(𝑥 + 𝑎) + 𝑠.

Therefore, when 𝑥 = −𝑎, it becomes 𝑝(−𝑎) = ℎ(−𝑎)(−𝑎 + 𝑎) + 𝑠.

So, 𝑝(−𝑎) = 0 + 𝑠. Thus, 𝑝(−𝑎) = 𝑠.

Because 𝑠 is the remainder, then the remainder is 𝑝(−𝑎).

So, if 𝑝(𝑥) is divided by (𝑥 + 𝑎), the remainder is 𝑝(−𝑎).

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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.

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