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An example of Proof-Based Teaching : 3rd graders constructing knowledge by proving
Estela Vallejo Vargas, Candy Montañez
To cite this version:
Estela Vallejo Vargas, Candy Montañez. An example of Proof-Based Teaching : 3rd graders con- structing knowledge by proving. CERME 9 - Ninth Congress of the European Society for Research in Mathematics Education, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic. pp.230-231. �hal-01281144�
230 CERME9 (2015) – TWG01
An example of Proof-Based Teaching: 3
rdgraders constructing knowledge by proving
Estela Vallejo Vargas and Candy Ordoñez Montañez
Pontificia Universidad Católica del Perú, Lima, e.vallejo@pucp.pe
This study provides an example of what proof-based teaching is and how students of elementary school lev- el can construct their own knowledge about division and divisibility of natural numbers by following this approach.
Keywords: Proof-based teaching, constructing mathematical knowledge, elementary school level mathematics.
BACKGROUND
In the last decades research on the teaching and learn- ing of mathematical proof has substantially increased (Blanton et al., 2011; Reid & Knipping, 2010; among oth- ers). In addition to this, there is worldwide a growing tendency to include mathematical proofs in school programs, including at the elementary level, as ex- emplified by the Principles and Standards for School Mathematics from the National Council of Teachers of Mathematics (NCTM, 2000).
Despite all of this interest, most of the time it is still quite common to find researches focused on mathe- matical proof as a subject of study and not as a means to contribute to constructing mathematical knowl- edge. On that subject, Reid (2011) has proposed that proving could be the vehicle for learning new math- ematics through what he calls “proof-based teaching”.
He tells us:
We must ensure that we see proof as fundamental to mathematics as a way to develop understanding of mathematical concepts, and as a way to discov- er new and significant mathematical knowledge.
Proof cannot be limited to the format of proofs, and to the role of verification of knowledge (for which there is probably good empirical or other evidence already). (p. 28)
THE EXAMPLE
The work of Ordoñez (2014) was developed with stu- dents around 7–8 years old who did not have prior knowledge about division when this research began.
This study provides a clear example of what Reid (2011) calls proof-based teaching. In this work, for which Estela Vallejo was the supervisor, Ordoñez shows how third graders are capable of constructing their own knowledge of division and divisibility of natural num- bers from the key notion of equitable and maximum distribution, which is understood by students in a natural way. The knowledge construction becomes evident when students are capable of answering prob- lems that demand justifications of their answers. In the process of knowledge construction, it can be seen that students not only participate actively, but are also encouraged to correct their classmates’ or their own answers, refine ideas, suggest conjectures, etc. All of this shows us that it is possible to develop a class- room environment rich in knowledge construction, in which the students experience similar processes to those experienced by professional mathematicians, including especially the process of proving to discov- er and establish new knowledge.
In this research study two important elements of proof-based teaching are combined: establishing a framework of established knowledge from which to prove, and establishing an expectation that answers should be justified within this framework.
This transcript from the class shows these two ele- ments:
Tutor: Can we have 3 marbles left after a dis- tribution of certain number of marbles among 3 people?
Student 1: No, because you have to distribute the maximum number of marbles.
An example of Proof-Based Teaching: 3rd graders constructing knowledge by proving (Estela Vallejo Vargas and Candy Ordoñez Montañez)
231 Tutor: So, does it mean I have not distributed
the maximum number of marbles?
Student 2: We can still distribute these 3 mar- bles! One more for each person!
The tutor’s question could be answered with a ‘yes’
or ‘no’, but the student provides a justification as well, in keeping with the expectation that answers should be justified. It refers explicitly to the basic notion of maximum distribution, which is part of the frame- work of established knowledge. The tutor questions whether this basic notion applies in this case, and the second student provides an additional justification, a backing for the use of the basic notion in this case.
This way of constructing division knowledge helped these students to realize why they cannot have 3, or a number greater than 3, as a remainder when they are dividing by 3.
REFERENCES
Stylianou, D.A., Blanton, M.L., & Knuth, E.J. (Eds.). (2009).
Teaching and learning proof across the grades: A K-16 perspective. New York: Routledge.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston VA: Author.
Ordoñez, C.C. (2014). La construcción de la noción de división y divisibilidad de números naturales, mediada por justifi- caciones, en alumnos de tercer grado de nivel primaria.
(Master’s Thesis). Retrieved from: http://tesis.pucp.edu.pe/
repositorio/handle/123456789/5653
Reid, D. (2011, October). Understanding proof and transforming teaching. In Wiest, L., & Lamberg, T. (Eds.), Plenary present- ed at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 15–30). Reno, NV: University of Nevada.
Reid, D., & Knipping, C. (2010). Proof in Mathematics Education.
Research, Learning and Teaching. Rotterdam, The Netherlands: Sense.
