Improving Reasoning Abilities of 5th-6th Grade Pupils Using a Specially Designed Teaching Unit in Pre - Formal Logic
Raisa Guberman and Marita Barabash
“… you should say what you mean,” the March
Hare went on.
“I do”,
“
L. Carrol,
1. The example in the first epigraph is a very typical one of the students
of vast range of ages, including some students in teachers’ education colleges,
who attribute no significance to the syntactic structure responsible for the
logical meaning of the phrase. They are sure that the words are the only
factors that matter, and are not aware of the change in meaning caused by the
change in the order of words. Thus, they are incapable for example to discern a
statement from its converse.
The second example is related to the way the
pupils arrive at conclusions, generalize and apply inductive arguments: since
it is very like Mabel to take
The two examples above illustrate some of
the phenomena related to the verbal expression of a
thinking process, in particular, of drawing conclusions.
2. Speaking of “
It is funny because it is funny: full
of folly and nonsense, mocking people and phenomena. But the fun is rather
sophisticated, being the fun of: a. a mathematician b. an English
mathematician. To enjoy the book a reader has to feel free with this
sophistication. The thinking processes and lingual skills needed for that are
far from being taken for granted for a reader. Hence, it is usually thought of
as a difficult book, especially if it is to be regarded a children’s book.
In the course of school teaching and learning
the skills mentioned above are not being sufficiently developed. This in turn
causes essential difficulties when a student arrives at the deductive stages of
mathematical learning, e.g. at the deductive geometry.
3.
What do we learn from these
and other examples from
·
Those
who have no experience in logically structured arguing do not see any need in
it, unlike those who, for example, don’t know the matrix calculus and are aware
of that. In short, they don’t know that they don’t know.
·
We are
sometimes unaware of the fact that we are being understood inversely to what we
have meant. Regarding teaching situations, this equally implies to a student
who sometimes means the opposite to what he says, and to a teacher who is
understood opposite to what he means and says, - even if what he means
is what he says, - and does not know that this may be the way he is understood.
·
Children
who master well enough basic arithmetic and geometric skills appropriate for
their age, don’t master basic reasoning skills, whereas the reasoning lingual
abilities are necessary in order for the mathematic learning to comprise
understanding and developing concepts, and not only to acquire skills.
·
Lingual
skills in young pupils are being developed with no connection whatsoever to
mathematics, specifically to logic. In particular, no emphasis is made on the
dependence of the meaning of a phrase on its syntactic structure. “I say what I
mean” and “I mean what I say” are two phrases that differ only in the order of
words, speaking at a basic level, or in the syntactic structure, speaking at a
more advanced level. At neither of these levels, the implication of this
difference for the meaning of the phrase is related to in any primary school
context. Actually, the pupils are not being taught to “weigh” the logical sense
of a phrase.
4. Students frequently ask, regarding i.e. a geometrical proof: is it all
right that it is written in words? This indicates that the rigor is related for
them to formal signs rather than to a formal structure behind the reasoning.
Thus, for example, they do not accept the proof by contradiction as reliable
enough because it is usually written in a lot of words; its formal logical
structure is not transparent for the pupils because of their lack of experience
in relating syntactic structures of verbal expressions to the logic.
5. Numerous researchers in the math education have referred to the need for
intermediate stages towards formal proof and reasoning in mathematics. In
particular, the idea of a pre-formal proof in various forms and implementations
has emerged and developed see e.g. Balacheff (1991), Blum
& Kirsch (1991), Pinto & Tall
(2002), Mariotti (1997), De Villiers
(1995), and many others. Various modifications of idea are based on visual
concepts of manual or computer origin. We assert that the idea of pre-formal
proof must include pre-formal logic. By this we mean a set of rules compatible
with the formal rules of classical deductive logic theory, but built in a
non-formal way using analogies, visual arguments etc., unlike the rigorous
logic theory based on truth tables,
predicate calculus etc.: “In our opinion, in teaching mathematics, thinking
in algorithms has an disadvantage over thinking logically. … We need to teach
logic in a different way, build upon the students existing logical thinking and
improve it by solving exercises” Bako (2002). It is possible to appeal to logic on the
basis of the common sense, similarly to how we appeal to students’ primary
notions in other fields acquired before the teaching, i.e. counting, figures
etc.
The pre-formal logic may be effectively
developed using a visual tool, which is the Venn diagrams, known also as Euler
diagrams.
6. Keeping in mind the purpose of
developing the pre-formal logic in primary school pupils, a group of math
educators from the Achva College have developed a
teaching unit named “Learning with Alice to Think and to Reason” (Barabash et.al., 2002) intended
for the pupils of 5th – 6th grades of primary school. It
will be described further in some detail. Based on this unit, we have planned
an experiment to assess the effect of teaching in thus designed logic
environment, on the development of pupils’ ability to reason logically and to
build logically valid argumentation.
We were interested in following research
questions:
·
What is the measure of pupils’ success in solving the problems requiring verbal reasoning and arguing proficiency?
·
Are the
children able to discern the logical difference between two statements
formulated in the same wording but of different syntactic structure?
·
What are the pupils’ strategies being used for assignments requiring reasoning?
Our hypothesis were:
·
There will be a difference in strategies used
by the pupils. The pupils from the experiment group will use strategies related
to the valid reasoning to a greater extent that those form the control group.
·
The
pupils from the experiment group will resort to the graphic representation tool
(Venn diagrams).
·
The
pupils from the experiment group will succeed better in solving problems
requiring valid reasoning, than those of the control group.
·
The
pupils from the experiment group will better solve “difficult” problems (i.e.
those requiring complicated multi-step reasoning) than those from the control
group. No significant difference is expected regarding “easy” problems.
·
The
visual tool combined with extensive verbal exercising will help the pupils to
discern the difference in phrases originating form different syntactic
structures, by inducing them to put their vague ideas into words, as precisely
as possible, orally and in the written form.
·
The use
of visual representation helps to solve “difficult “problems.
The research population:
Pupils of 5th
and 6th grades from two schools in the south of
The research tools:
1. The learning unit “Learning with
Part 1:
Graphical representation of statements:
acquaintance with the tool.
Part 2:
Constructing a statement: implication; the notion of implication and
Venn diagrams representing implications.
Part 3: Different statements – different
diagrams: various types of logical statements and
corresponding Venn diagrams.
Part 4: Validation of statements: using
appropriate Venn diagrams to validate statements.
The learning unit was developed according to
the following principles:
·
Development
of reasoning verbal abilities in mathematical and non-mathematical contents.
Pupils having problems with some mathematical topics or skills may cope
successfully with assignments based on non-mathematical contents, and thus
develop their inductive and deductive skills independently of their
mathematical background. All the pupils may use analogies in transferring their
common-sense valid argumentation in non-mathematical familiar contents into
logically similar mathematical situations.
·
Usage
of visual-graphic representation is expected to help to arrive at a decision
concerning the correctness of a statement.
·
As
numerous studies indicate (e.g. Anderson 1994, Chazan
(1993) and many others), pupils of all ages frequently have difficulties in
understanding logic relations between mathematical concepts. To overcome this
difficulty, we need to develop a mode of meaningful learning of these relations
on the basis of one’s previous experience and knowledge. We believe that this
approach will lead to the development of mathematical intuition based on valid
logical reasoning.
·
Each
series of assignments includes the logical analysis of at least one paragraph form the book
2. The research tools for
assessment of the impact of the teaching unit.
·
The
pre-test.
In order to answer the research questions, the test were comprised that
included various statements, some of them mathematical and some of them – of
non-mathematical contents related to the everyday life. The questions were of
two levels: “easy” ones and “difficult” ones. The “easy” questions were taken
from the usual repertoire familiar to a pupil (e.g. relations between
quadrilaterals or division properties of numbers). The “difficult” questions
were similar in wording to the ”easy” ones, but contradicted in some way or
other the intuitive models of the phenomena built up by the pupils. The third
type of question was an “applied” type of question: logical formulation of an
everyday situation.
·
The
post-test: the same.
Results:
1.
The
pupils from the experiment group used the Venn diagrams to solve the questions
of the post-test. This tool appeared to be useful to them in formulating
successfully valid arguments and explanations.
2.
The
level of success in problem solving was significantly higher in the experiment
group than in the control group.
3.
No
significant difference was discovered between the success
in problem solving as far as “easy” problems were concerned1. As to the “difficult” and applied
problems, the pupils from the experiment group succeeded significantly better
that those from the control group.
4.
Those
pupils from the experiment group who used the Venn diagrams to solve the
“difficult” problems, succeeded better than those who did not use them.
7. Conclusions
·
In
designing the logic learning environment, it is advisable to base upon two
components: a. A content component, e.g. a children’s book; b.
Visual-graphic tool – Venn diagrams. These two components serve as intermediaries
in the learning process.
·
The
usage of the graphic tool helps a pupil to recognize the same logic structure
in statements different in their content, and thus to check their validity
using logically based correct analogy.
·
The
experience acquired thanks to the teaching unit enhances the ability of a pupil
to cope with “difficult” and “applied”
problems by accumulating a variety of tools useful in problems requiring
argumentation and reasoning.
·
The
pupils became more sensitive to the structure of a phrase, and did their best
to analyze it. In doing so, they resorted to the Venn diagrams as to a reliable
tool.
8. Discussion
One cannot ignore
the need to develop skills needed for valid mathematical reasoning, along with
other aspects of mathematical knowledge, though it is far from simple a
task:
“Teaching students to both recognize and produce valid mathematical arguments is certainly a challenge. We know all too well that many students have difficulty following any sort of logical argument…. We cannot avoid this challenge, however. We need to find ways, through research and classroom experience, to help students master the skills and gain the understanding they need.” (Hanna 1996). See also ” The teaching of proof should not only focus on the product … but also (and even much more) on the learning of the process… That process is complex and difficult enough … it is the responsibility of the teacher to guide the students into this difficult and long term learning…”. (Douek 2000).
We assert that some systematic teaching of logical skills will improve significantly the learning process in mathematics. Inductive and deductive abilities are important both on a macro level, i.e. in the development of a pupil’s math perception, and on a micro level, i.e. in any unfamiliar mathematical situation (see e.g. Polya 1954). It is undoubtedly a complex task, in particular because a math teacher has to attend to lingual skills as well. This complexity may be the reason that not much attempts of teaching logic at any level are undertaken at school. We suggest that this is both essential and possible.
Bibliography
1.
2.
Bakó M.(2002) ‘Why we need to teach logic and
how can we teach it?’, http://www.ex.ac.uk/cimt/ijmtl/bakom.pdf.
3.
Balacheff N.(1991) ‘The benefits and limits of social interaction:
the case of mathematical proof’, in A. Bishop, S. Mellin-Oi.Sen
& J. Van Dormolen (Eds)
Mathematical Knowledge: its growth through leaching (
4.
Barabash M., Osviazov H., Shamash J., Guberman
R., Shimanovich M.(2002),
‘Learning with
5.
Blum W., Kirsch A.(1991).
‘Preformal Proving: Examples and Reflections’, Educatinal Studies in Mathematics 22, 183-203
6.
Carrol L. (1970) The Annotated
7.
Chazan D.(1993). ‘High School Geometry Students’ Justification for
Their Views of Empirical Evidence and Mathematical Proof’, Educational
Studies in Mathematics 24,:359-387
9. Douek N. (2000).’Comparing argumentation
and proof in a mathematics education perspective’. Contribution to: Paolo Boero,
G. Harel, C. Maher, M. Miyazaki (organisers)
Proof and Proving in Mathematics Education. ICME 9
TSG 12. Tokyo/Makuhari,
12. Mason, J., Pimm,
D. (1984) ‘Generic Examples: seeing the general in the particular’, Educational
Studies in Mathematics, 9(2), 2-8.
Notes:
1 This supports our presumption of sufficient
basic common sense of the pupils of this age to base upon.