Education Reform Journal - Volume 5 - Number 1 - 2020 - Examining the Processes of High School Students to Do Proof Without Words

Oct 15, 2026

Opinions in mathematics education highlight the need to shift from training formal proving skills toward helping students understand mathematical proofs. This study investigates how high school students in Turkey carry out proofs without words, described as informal proof approaches. Designed as a case study, the research emphasizes activities that position students actively in the proof process. Teachers are expected to guide learners by being aware of proof process stages.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages19

Reading Time48 min

AccessFree to download

Abstract

Introduction

Proof as a key mathematics practice

Proof without words and its use in teaching

Mental activities in proof processes

Guiding students through proof stages

Figure 1. Examples of proof without words

Education Reform Journal Volume 5 Number 1, 2020 e-ISSN: 2602-3997 [http://dx.doi.org/10.22596/erj2020.05.01.8.26](http://dx.doi.org/10.22596/erj2020.05.01.8.26) Examining the Processes of High School Students to Do Proof Without Words* Kubra Polat 1 LeventAkgun2 Submitted: June 19, 2019; Accepted: June 18, 2020; Published Online: June 29, 2020 Abstract There are opinions about approaching the proof in mathematics education need to change as providing students to understand the mathematical proof rather than developing formal mathematical proof skills. In this context, proofs without words described as informal proofs can be used in proof teaching. The purpose of this study was to investigate the processes of high school students to do proofs without words. This study was designed as the case study a high school in Turkey. Consequently, activities of proof without words that allow the student to take an active role in the proof process can be presented as an alternative method in proof teaching and the proof without words can be used for teaching students the stages of proof process. It is necessary for teachers to be aware of the proof process stages and guide the students. Keywords: Proof without words, visual proof, proof process, Boero’s model Introduction Proof is considered as the highest level of mathematics practice (Miller, Infante, & Weber, 2018; Tall, Yevdokimov, Koichu, Whiteley, Kondratieva, & Cheng, 2011) and many mathematicians criticise the proof teaching approaches presented in the schools and the studies have tried to reveal new ways for the proof teaching and learning (Hanna, 2000; Hanna & de Villiers, 2012) . One of the main research topics of the mathematics is to determine whether or not the mathematical proof is used to justify mathematical propositions by the students and to what extent they are used (Rodd, 2000) . Indeed, the proof is a ritual made by many students without understanding and the studies have showed that students cannot even do and understand simple proofs (Ball, Hoyles, Jahnke & Movshovitz-Hadar, 2002; Harel & Sowder, 2007; İnam, Ugurel & Yaman, 2018) . Although the key role of the proof is to increase the mathematical understanding, the real challenge is to find the most effective ways of using proof for this purpose (Hanna, 2000) . It is important * This study was produced from doctoral thesis titled―Proof Without Words as an Alternative Proof Method: Investigating High School Students’ Proof Skills‖ * The part of this study was presented as an oral presentation at UFBMEK 2018 (13th National Congress of Science and Mathematics Education) (UFBMEK, 2018), Denizli, Turkey. 1Sivas Cumhuriyet University, [0000-0001-8060-0732], [kubraapolaat@hotmail.com.tr](kubraapolaat@hotmail.com.tr)[ ](kubraapolaat@hotmail.com.tr)2Ataturk University, [0000-0002-1435-1771], [levakgun@gmail.com](levakgun@gmail.com) to show the proof to the students as a way of showing the accuracy of mathematical expressions to other people and helping them to understand why the theorem is correct so that the students understand the proof and comprehend its importance (Alibert & Thomas, 1991) . According to Ugurel, Morali, Karahan & Boz (2016) if the proof is ensured to be understood by students, it becomes possible to change their negative views about the proof. It is an important skill to understand a proof, but it is not necessary for the student to have a high mathematical success for understanding the proof. So we can state that, it is necessary to provide an opportunity to the students for understanding the proof. Proof requires a series of mental activities such as research, the use of mathematical concept knowledge, ability to apply heuristic strategies to the problem, determining the hypotheses, organising logical arguments, analysing them, and organising the necessary transformations. The reason for the failure in proof teaching is the lack of recognising this complex process. Un

FAQs

What is the main purpose of the study?

To investigate the processes of high school students to do proofs without words and to consider it as an alternative method in proof teaching.

How are proofs without words described in the document?

They are informal proofs that use visual presentations to present a mathematical idea, equation, or theorem, relying on geometric drawings and symbols rather than words.

What role do teachers play according to the study?

Teachers should be aware of the stages of the proof process and guide students during proof teaching so learners can understand each stage.

Oct 15, 2026

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