Some Primary-School
Teachers’ Conceptions on the Mathematical Notion of Volume
|
Mariana Sáiz Universidad
Pedagógica Nacional Mexico |
Olimpia Figueras
Cinvestav, IPN,
Mexico dfiguera@mailer.main.conacyt.mx |
The concept of volume is one of
the topics in the Mexican curriculum of basic education, and serving
primary-school teachers have pointed out that this is one of the themes where
their students face difficulties. On
the other hand, although the volume mental object is quite a wide issue
(Freudenthal, 1983), there is evidence that teachers limit themselves to relate
it to the notion of capacity and with its calculation by means of
formulas. This situation motivated the
creation of a research project which inquired into the teachers’ ideas on this
mathematical concept and its teaching;
a part of the findings are reported herein.
|
Questionnaire |
Type of data |
Examples of questions posed |
|
1 |
The participants’ professional profile and work
experience. |
In what grade have you worked the most during your
years in charge of a group? |
|
2 |
The teacher’s ideas about the teaching of
mathematics in general, and of volume in particular. Among other things, there is an interest
in finding out whether the teachers are using the bibliographic material
provided by the Ministry of Public Education, and how deeply they have
studied it, especially concerning the part related to volume. On the other hand, hypothetical situations
are proposed to the teachers with the intent that they evaluate these
situations dealing with the learning of volume and other volume-related
situations. |
1. Teacher Julián asked his
students to calculate the volume of a cup. a) Amalia poured water into the
cup and then measured the liquid by means of a graduated container. What was it that Amalia measured? b) Ana put the cup inside a graduated
vessel containing water, and measured how much the water-level rised. What did Ana measure? c) Which of the two girls measured the
volume of the cup? 2. Julio, a sixth-grade
student, was asked by his teacher to calculate the volume of a parallelepiped
made up of cubes (as shown in a figure that contains a picture of a 2 ´ 4 ´ 8
parallelepiped). Julio’s answer was:
“the volume is 64, because I can undo the parallelepiped and, with the same
number of small cubes, I can build a cube measuring 4 on each side. I know that to calculate the volume of a
cube you multiply 4 ´ 4 ´ 4, and when I solve this operation the result is
64”. If Julio were your student and
he answered to you the way he answered his techer, a) what would you say to
him? b) Why? c) Would you recommend to him to do things
differently? If so, what procedure
would you suggest to him? If not,
why? |
|
3 |
The teacher’s ideas on the mathematical
notion of volume, such as the various meanings attributed to that word, and
the properties of bodies in relation to volume. |
Considering the following list
of properties of bodies, answer “yes” or “no” depending on whether you
believe that the property in question is, or is not, related to volume. Finally, comment on why you say “yes” or
“no”. 1) 1)
mass 2) temperature 3) lateral area 4) capacity 5) weight |
Taking into
account that the written answers do not provide a complete information about
the teachers’ ideas and knowldge (Thompson 1992), a workshop was designed where
the teachers could spontaneously comment and share their ideas with their peers
when working with tasks related to the concept of volume. The workshop was presented as an instrument
for data collection, and not with the purpose of assessing the impact it could
have on the teachers’ knowledge and ideas, even if one is aware that some
learning or a new conceptualization might occur. A description of the structure and the topics dealt with in a
workshop with 24 teachers, in five sessions, each lasting four hours, is
presented in Table 2.
Session
|
Description |
|
1 |
Tasks on spatial imagination
and the estimation of capacities without the use of conventional units. Review of textbooks in order to identify
the lessons related to volume. |
|
2 |
Activities dealing with
relationships between volume and capacity, and problems related to the
conversion of capacity and volume
units. |
|
3 |
Activities permitting a
reflexion on methods to calculate or to compare the volumes of irregular
bodies, and the variations between volume and lateral area, and viceversa. |
|
4 |
Analyses on the effect on a
body of lengthening or shortening the linear magnitudes of such body. |
|
5 |
The construction of bodies, and
the make-break transformations to calculate and compare volumes, and to
deduct formulas for geometrical solids.
The analysis of the formulas used in grade-school. A reflexion on the reading of some
historical passages related to the origin of some formulas. |
The data were
subjected to iterated scrutinies.
During the first of these scrutinies, evidences were identified and
classified on the uses of the teachers’ mathematical knowledge on the topic,
and on their beliefs about mathematical teaching in general, in their written
answers and their verbal comments. In
the next stage, the classes obtained in that first scrutiny were constrasted
with the conceptual network, in order to identify the uses of qualitative and
quantitative aspects that teachers make regarding the general aspects of the
mental object volume as characterized in such a network. The purpose of the next stage was to
identify those elements of the mental object which were tied to the
mathematical properties closely linked to the volume mathematical concept from
the standpoint of the formal component.
Discussion and findings
From
the present stage of the analysis of the great amount of data derived from the
work with the teachers in the workshop, three findings have been selected which
are considered to be significant for the present article; these will be
presented in the following paragraphs.
1) The belief in the existence of a
dependency relationship between the lateral area and the volume. In order to collect information on this
issue, a question in the questionnaire No. 3 which is shown on Table 1 was
designed. Nine of the twelve teachers answering the question stated that the
lateral area and the volume are related,
and the arguments of four of them agreed in pointing out that: it is determined by the dimensions which,
in turn, determine its capacity or
volume. This evidence shows a tendency among the participants to consider a
dependency between the lateral area and the volume, a fact that appeared later
on, during the third workshop session (see Table 2) whose purpose was precisely
to provoke a discussion about this belief.
In Table 3, the extract of the transcription illustrates the fact that
problems such as the one shown, and the ways in which teachers solve them, tend
to strengthen such a belief.
|
Problem |
Transcription of the
discussion |
|
A factory produces water tanks by means of rectangular steel sheets; if it is intended that such tanks have a maximum capacity, how is it convenient to roll the sheets, lengthwise, along their width, or is it indiferent? |
Researcher: Did you think initially that one was bigger, or that they were both equally big? Teacher No. 1: Here we even established that they were both equal; because we said to ourselves, look, if the sheet is like this and I fold it (he rolls a sheet of paper along its width) like this, or if I fold it like this (he rolls it lengthwise) the result is the same, because the area does not vary. |
Table 3 the problem of the water tanks and the
following discussion
It should be mentioned that
in order to solve that problem all the teams used a particular case, a rectangle
5 x 10 cm, from which they derived a general rule. In three of the teams a calculation was performed using those
numbers, and in one, work was done with paper models and the pouring of seeds. Upon facing the answers to the problem, and
in spite of the fact that they have realized that the volumes of the
constructed and calculated cylinders are different, and that the workshop
conductor focuses attention on the equality of the lateral areas, the teachers
argued that the area of the base was missing, and that that was the reason for
the difference in the volumes. The
workshop included other activities where it became clearer that there can be
two bodies with the same volume and a different lateral area, and work was done
on these later on.
2) The association of the change in the level of a liquid with the
weight of the body when submerging it in a vessel containing water. In order to find evidence on
this belief, question 1 from the questionnaire 2 was posed (see Table 1). Even
when in the questionnaire only one teacher, from thirteen, associated the
weight with the change in the level of water, during the third workshop
session, where it was suggested that a comparison is made of the volume of
irregular bodies in order to evoke the use of the Archimedes’ principle, the
trend favoring this belief manifested itself more vigourously. The extract from
a discussion during the team-work, shown in Table 4, where certain phrases have
been emphasized by the use of bold-face, comes to prove our former statement.
|
Researcher: [...] Do we all agree that we are measuring volume? A chorus of the whole group: Yes. Teacher No. 2 (from one
team): It is volume. In the experiments we performed we saw
that volume was actually measured; we concluded that volume is the place that
a body occupies in space. In order to
prove this –considering that there were some doubts among our colleagues—we
took one of these cubes (takes a plastic cube which can be opened and closed)
and we filled it with modelling clay; it
was heavier, it sank. The one
that was empty, we had to push with a pencil; in other words, weight helps
immersion. Yet, in the level of water with modelling clay and with the empty,
closed cube, the same displacement occurred.
That means that weight has nothing to do with displacement. Teacher
No. 3 (from another team): You mean
that with modelling clay inside, or
without it, the rise is the same? Teacher
No. 2 and his team in chorus: Yes. |
Table 4 Extract of a discussion around Archimedes’ principle
The fact that they had
designed the experiment of filling a plastic cube in order to have two bodies
with identical volumes but with different weights, to see if this did not
affect the change in level, shows that more than one of the teachers had this
doubt; and this is corroborated when a teacher from yet another team asked,
incredulous, whether the level of water had risen in the same measure.
3) A qualitative didactic approach, versus
the use of formulas. One final
aspect which it was seen fit to explore had to do with the teacher’s ideas concerning the teaching of
volume. In order to collect information
on this point question No. 2 in questionnaire No. 2 was designed (see Table 1). Out of nine participants who answered items
b) and c) of this question, three of them stated that they would not suggest to
Julio a different way of doing things, and they argued in the sense that
children must use their own
procedures. The written responses
are now given of those teachers who answered in the affirmative, and the way they
argued their opinions. Teacher No. 8: Yes, the application of the formula V = the area of the base x the height; but first we would have to perform activities leading to the
comprehension of this measurement.
Teacher No. 15: Yes, to measure
what the base contains (how many cubes will fit in it) and then multiply this
by the number of “floors” [i.e., the number of levels] that are repeated.
Other teachers
tried to direct the children towards a more profound relationship between the
linear dimensions of the body and its volume, and some of them intended for the
child to deduce such a relationship. Teacher No. 2: Let him/her leave four cubes as a base. Teacher No. 10: Yes, that
he should try to find some regularity akin to the one he/she found for the
cube, in order to obtain the volume of a parallelepiped. Teacher No. 6: Yes, I would ask him/her: How would you
calculate the volume without breaking it up.
Imagine that I cannot dismount it because the cubes are glued.
As it can be
seen, a majority tendence made itself evident towards a valuation of the
quantitative aspect in the study of the notion of volume; the use of the make
break transformations that Freudenthal and other researchers consider
fundamental for the formation of the volume mental object was not valued. This trend also arose during the workshop;
to illustrate this episode, a dialogue is included taken from the transcription
pertaining to the first session, where –among other things--, the textbooks
were examined –these being, as it will be remembered, of nation-wide
application, the only ones, and mandatory—where it is intended to develop a
teaching model for volume through a primarily qualitative approach. Teacher No. 5: Oh!
Here, even a problem on how to measure is shown, [...] how to deliver
12
liters [...] we say that this does pertain to volume. [...later, about a different lesson] yes,
the fact is handled that the box is filled up.
But not because a box is filled this must pertain to volume; that’s
the way we feel about it. These kinds of expressions are fairly abundant in
all the workshop sessions, including those dealing with spatial imagination,
where the calculation of volumes is not involved.
Concluding
Remarks
In
this article, three trends have been discussed about the teachers’ beliefs: two
of them have to do with their knowledge about the concept of volume, and the
other falls into the sphere of the teaching of such a notion. One aspect which
deserves greater attention, and perhaps further studies is the one linked to
the difficulty of changing some of the teachers’ beliefs, as we have already
said. The problems commonly posed to
generate a discussion on such beliefs, and the methods the teachers use to
solve the problems –such as generalizing particular cases– only succeed in
reinforcing the aforesaid beliefs.
Another aspect which deserves reflexion is that a reform was carried out
in Mexico, which included changes in the curricula and in the approach to the
educational proposal for primary-school education; it also included the
preparation of new didactic materials such as textbooks, activity files, and
teachers' books which reflect such changes.
These materials have been used since 1993, and they were preceded by
workshops and courses for the teachers who worked with them. Throughout this
process, a vision of mathematics is proposed where many topics are approached
with a qualitative focus, as in the case of volume. A fact that has been confirmed by the present investigation,
however, is that this effort becomes clouded by the value that many teachers
confer to the quantitative aspects.
This situation highlights the need for studies leading to strategies and
situations which may facilitate a change in the teachers’ conceptions about
what teaching, learning and doing mathematics might mean.
References
Filloy,
E. (1999). Aspectos Teóricos del Álgebra
Educativa. México: Grupo Editorial Iberoamérica.
Freudenthal, H. (1983). Didactical
Phenomenology of Mathematical Structures, Holland: Reidel Pub. Co.
Sáiz,
M. (1998). How has measurement been taught in Mexico? Proceedings of the XXth Annual Meeting of the PME-NA, North
Carolina, pp. 328-334.
Sáiz, M. and Figueras, O.
(1999). A conceptual network for the teaching and learning processes of the
mathematical concept volume. Proceedings
of the XXIth Annual Meeting of the PME-NA, Cuernavaca, Mor. 436-442
Thompson, A. (1992).
Teacher’s Beliefs and Conceptions: A synthesis of research. In Douglas A. Grow
(ed) Handbook of Research on Mathematics
Teaching and Learning. NY: Macmillan Pub. Co.