ICME Monographs. Guedet Eds. Classroom-based interventions in the area of proof: Some design considerations. Hamburg, Germany. Li, H-C. The roles of teacher and students during a problem-based learning intervention. Czech Republic, Prague. Algebra-related tasks in primary school textbooks. Nicol, P. Liljedahl, S. Allan Eds. Vancouver, Canada. If not textbooks, then what? Jones, C. Bokhove, G. Fan Eds. Southampton, UK: University of Southampton. Seoul, Korea. Pytlak, T.

Swoboda Eds. France, Lyon. Lin, F. Hsieh, G. The mathematical preparation of teachers: A focus on tasks. Monterrey, Mexico. San Diego, California. The mental models theory of deductive reasoning: Implications for proof instruction. Philippou Eds. Cyprus, Larnaca. Content knowledge for mathematics teaching: The case of reasoning and proving. Prague, Czech Republic.

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Alatorre, J. Cortina, M. Vincent Eds. Melbourne, Australia. Reconsidering the drag test as criterion of validation for solutions of construction problems in dynamic geometry environments. Denmark, Copenhagen. Triantafyllides, K. Hadjikyriakou, P. Chronaki Eds. University of Thessaly, Volos, Greece. Tzekaki Ed. Aristotle University of Thessaloniki, Thessaloniki, Greece. B, pp. The Society, Nicosia, Cyprus. The transition from informal to formal proof.

Makrides Eds. Cyprus Mathematical Society, Larnaca, Cyprus. Search site. International students Continuing education Executive and professional education Courses in education. Research at Cambridge. Andreas Stylianides. Eleni Demosthenous : Algebra-related topics: A multiple case study in Cypriot primary school classrooms. Research project with M. Jamnik funding opportunity available Supporting secondary mathematics teachers to enhance their students' proof competencies.

Research project with G. Research project with H. Role: Principal Investigator. Numbered List of Outputs referenced in the End of Award Report ; Numbered List of Outputs referenced in the Impact Report completed Preservice teachers' challenges in beginning to teach mathematics: The activity of reasoning and proving.

Research project funded by the Spencer Foundation. Role: Co-principal Investigator with G. Stylianides and A. Chapters in Research Handbooks Stylianides, G. A particular way of organising a proof using two parallel columns is often used in elementary geometry classes in the United States.

In each line, the left-hand column contains a proposition, while the right-hand column contains a brief explanation of how the corresponding proposition in the left-hand column is either an axiom, a hypothesis, or can be logically derived from previous propositions. The left-hand column is typically headed "Statements" and the right-hand column is typically headed "Reasons".

The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with mathematical objects , such as numbers, to demonstrate something about everyday life, or when data used in an argument is numerical. It is sometimes also used to mean a "statistical proof" below , especially when used to argue from data. While using mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that the assumptions from which probability statements are derived require empirical evidence from outside mathematics to verify.

In physics , in addition to statistical methods, "statistical proof" can refer to the specialized mathematical methods of physics applied to analyze data in a particle physics experiment or observational study in physical cosmology. Proofs using inductive logic , while considered mathematical in nature, seek to establish propositions with a degree of certainty, which acts in a similar manner to probability , and may be less than full certainty. Inductive logic should not be confused with mathematical induction.

Bayesian analysis uses Bayes' theorem to update a person's assessment of likelihoods of hypotheses when new evidence or information is acquired. Psychologism views mathematical proofs as psychological or mental objects. Mathematician philosophers , such as Leibniz , Frege , and Carnap have variously criticized this view and attempted to develop a semantics for what they considered to be the language of thought , whereby standards of mathematical proof might be applied to empirical science.

Philosopher-mathematicians such as Spinoza have attempted to formulate philosophical arguments in an axiomatic manner, whereby mathematical proof standards could be applied to argumentation in general philosophy. Other mathematician-philosophers have tried to use standards of mathematical proof and reason, without empiricism, to arrive at statements outside of mathematics, but having the certainty of propositions deduced in a mathematical proof, such as Descartes ' cogito argument. Sometimes, the abbreviation "Q. This abbreviation stands for "Quod Erat Demonstrandum" , which is Latin for "that which was to be demonstrated".

From Wikipedia, the free encyclopedia. See also: History of logic. Main article: Direct proof. Main article: Mathematical induction.

## Educational Research: Proofs, Arguments, and Other Reasonings

Main article: Contraposition. Main article: Proof by contradiction. Main article: Proof by construction. Main article: Proof by exhaustion. Main article: Probabilistic method.

### Curriculum, Instruction, and Pedagogy ARTICLE

Main article: Combinatorial proof. Main article: Nonconstructive proof. Main article: Statistical proof. Main article: Computer-assisted proof. Main article: Experimental mathematics. Animated visual proof for the Pythagorean theorem by rearrangement. Main article: Elementary proof. Main articles: Inductive logic and Bayesian analysis. Main articles: Psychologism and Language of thought. Main article: Q. Logic portal Mathematics portal. University of British Columbia. Retrieved September 26, A statement whose truth is either to be taken as self-evident or to be assumed.

Certain areas of mathematics involve choosing a set of axioms and discovering what results can be derived from them, providing proofs for the theorems that are obtained. The Nuts and Bolts of Proofs. Academic Press, Discrete Mathematics with Proof. Definition 3.

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Cambridge University Press. The development of logic New ed. Oxford University Press. January []. No work, except The Bible, has been more widely used See in particular p. Two thoughts surfaced: the first was that they were unpublishable in the standard way, there were no theorems only very suggestive pictures. The shift needs to pass through Level P1, C1 to enhance planning and constructing a proof reciprocally.

The transition process of the former shift passes through either Level C1 or P1. In the case of passing through C1, this level aims for students to connect assumptions and conclusions by a chain of singular propositions with hypothetical syllogism. Then, at the next level P1, C1 , the learning of P1, that is, clarifying what can be used and how it can be used to connect premises and conclusions, can be realized. In contrast, in the case of passing through Level P1, the learning of P1 cannot be realized because students cannot learn to clarify what can be used and how it can be used to connect the premises and conclusion P1 without having a chance to form and express the deductive connection between them C1.

Therefore, they first understand this connection in the Level C1 proof Figure 3 , and form and express by themselves by focusing on hypothetical syllogism. At that time, they might be able to clarify what can be used and how to connect the premises and conclusion of new proof problems. In passing through P1, C2 , this level aims to form and express the connection between the premises and conclusion with differentiating universal instantiation and hypothetical syllogism from deductive reasoning.

Due to carrying out deductive reasoning based on universal instantiation, at the next Level P2, C2 , thinking backward from conclusions and forward from assumptions can be differentiated and carried out together. However, in passing through Level P2, C1 , the learning of P2 is difficult to realize because students cannot learn to distinguish thinking forward from the conclusion, from thinking forward from the premises, and how to connect these processes P2 without the chance to form and express the deductive connection based on universal instantiation and hypothetical syllogism C2.

To think backward from the conclusion, students first must understand the proof structure with differentiating universal instantiation and hypothetical syllogism in the Level C2 proof Figure 3 , and form and express by themselves by focusing on universal instantiation Miyazaki et al. At that time, even to solve a new proof problem, they might be able to consider how to think backward from the conclusion and forward from the premises, as well as how to connect them.

Concerning Question a , we could specify the six learning levels and establish the two transition processes as learning progressions. For each level, the looking back component [Examining, Improving, and Advancing EIA ] should be encouraged in explorative proving Figure 4. Depending on the objectives, it should be decided whether looking back EIA is required or not. Therefore, by regarding these learning progression transitions, we can theoretically establish the following progressive model as a framework for developing an explorative proving curriculum:.

To answer Question b , we first examined the existing intended curriculum and lessons in Japan's Course of Study, to show how we can make them more explorative based on our theoretical framework. In lower high school in Japan, the Course of Study requires students to learn about the various properties of plane and three-dimensional figures mainly based on congruency and similarity, as well as the meaning of proof and how to prove formally.

Although the Course of Study encourages teachers to introduce formal proofs gradually until the end of Grade 8 14 years old , previous studies have proposed no clear method. To improve this situation, we first considered the correspondence of the intended units in the Course of Study with the two learning progressions in our theoretical framework, as illustrated in Figure 4. For example, in Grade 8 geometry, the Course of Study requires students to study the following units: the properties of parallel lines and angles, the properties of angles of polygons, congruence and the properties of congruent triangles, proof and proving, and the properties of triangles and quadrilaterals.

Table 1. Correspondence of intended units with local progressions in Grade 8 geometry. We have been designing and implementing junior high school lessons in our correspondence table derived from our theoretical examinations described above since We used the method of lesson study Lewis et al.

## Educational Research: Proofs, Arguments, and Other Reasonings | Paul Smeyers | Springer

Every teaching unit in the Course of Study includes many learning objectives. To realize lessons based on the correspondence table Table 1 , it is necessary to localize each pair with the unit contents. Six aim to teach students about the essential geometrical properties and apply the learned properties. In preparing the series of lessons, we worked with expert mathematics teachers to design the local progressions. We then designed the following local progressions. This school is in the city center and has six Grade 8 classes.

At that time, the teacher had 15 years of experience teaching junior high mathematics. Along with the plan, the teacher prepared lessons. After implementing each lesson, the teacher and researchers reflected on the implemented lessons, and refined the next lesson plan. According to the local progressions in the unit, the first to fourth lessons aimed to shift the learning level from P1, C1 to P1, C2. Concerning P1 in every lesson, the teacher always asked students how to solve the problem from the perspective of what could be used and how. Conversely, the shift from C1 to C2 must differentiate universal instantiation and hypothetical syllogism from deductive reasoning.

Before Lesson 4, the six properties of angles in lines were found inductively, proved using simple deductive reasoning, and shared as theorems. Figure 5. Problem and students' explanations with the teacher's comments in yellow chalk. This only showed singular propositions peculiar to the diagram and a chain of them using hypothetical syllogism. This suggestion encouraged students to show not only how to connect singular propositions with hypothetical syllogism in their proofs but also which universal propositions were necessary to deduce singular propositions from universal instantiations.

Next, the teacher selected two other students' explanations. After they finished writing their explanations, respectively, the teacher interacted with them as follows:. S1: I think the properties used are necessary to calculate angles as reasons, as we learned. T: The same, S2? T: Great! Thus, the teacher praised these students for correctly embedding the numbers of the theorems e. Through praise, the teacher again encouraged the students to indicate which universal propositions theorems were necessary to deduce singular propositions from universal instantiations.

This is a typical problem in the traditional curriculum. However, in the explanation, each calculation expression is drawn by universal instantiation using a geometrical theorem, and the order of applying theorems shows the deductive connection between the premises and conclusion by hypothetical syllogism.

## Mathematical proof

Therefore, it is clear that most students' ability could reach Level C2 proof by the implemented curriculum. By the reflections, we refined the correspondence of intended units with local progression Table 1. Although gradual shifts in producing proofs in curriculum have been recognized as important, it can be realized by introducing local progressions corresponding to unit learning objectives based on the theoretical framework of explorative proving.

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Moreover, our curriculum will encourage students to plan, construct, and look back at their proof and proving. In particular, the learning levels of looking back provide opportunities to examine, improve, and advance their proving activities more creatively and logically. Our idea for realizing a gradual shift is rooted in hypothetical learning trajectory Clements and Sarama, and learning progression Empsom, The theoretical process of curriculum development can thus proceed as follows: identifying the key components of the learning objectives, setting their gradual levels, respectively, combining them, elucidating the target progressions, and connecting the units with the progressions.

Furthermore, with adopting the method of lesson study Lewis et al. By implementing and reviewing these lessons, we could find the advantages and limitations of the curriculum, and proceed with developing a more robust curriculum by refining the correspondence of intended units with local progressions. We will continue to develop and improve the curriculum by reconstructing the theoretical framework by refining the target progressions, and adjusting the corresponding unit tables, to ensure that both theoretical and realistic dimensions are considered, thereby ensuring that it is practical and valuable for teachers Davis et al.

Concerning our research activity, we have already made the correspondence tables of units with our target progressions for Grades 7—9 geometry along with the Japanese Course of Study. Under the fruitful collaboration with expert teachers, these tables will be subdivided into local progressions according to unit learning contents, and then lesson planning and implementation will also proceed.