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
constructing knowledge by proving. Konrad Krainer; Naďa Vondrová. CERME 9 - Ninth
Congress of the European Society for Research in Mathematics Education, Feb 2015, Prague,
Czech Republic. pp.230-231, Proceedings of the Ninth Congress of the European Society for
Research in Mathematics Education.
HAL Id: hal-01281144
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An example of Proof-Based Teaching: 3rd graders
constructing knowledge by proving
Estela Vallejo Vargas and Candy Ordoñez Montañez
Pontificia Universidad Católica del Perú, Lima, [email protected]
This study provides an example of what proof-based
teaching is and how students of elementary school level 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 learning of mathematical proof has substantially increased
(Blanton et al., 2011; Reid & Knipping, 2010; among others). In addition to this, there is worldwide a growing
tendency to include mathematical proofs in school
programs, including at the elementary level, as exemplified 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 mathematical proof as a subject of study and not as a means
to contribute to constructing mathematical knowledge. On that subject, Reid (2011) has proposed that
proving could be the vehicle for learning new mathematics 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 discover 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)
CERME9 (2015) – TWG01
THE EXAMPLE
The work of Ordoñez (2014) was developed with students 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 numbers 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 problems 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 classroom 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 discover 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 elements:
Tutor:
Can we have 3 marbles left after a distribution of certain number of marbles
among 3 people?
Student 1: No, because you have to distribute the
maximum number of marbles.
230
An example of Proof-Based Teaching: 3rd graders constructing knowledge by proving (Estela Vallejo Vargas and Candy Ordoñez Montañez)
Tutor: So, does it mean I have not distributed
the maximum number of marbles?
Student 2: We can still distribute these 3 marbles! 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 framework 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 justificaciones, 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 presented 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.
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