ED6639
Technology and the teaching and learning of mathematics.
Winter 2008 with
Margo Kondratieva.
Meeting 4. on February 5
Problems on geometrical constructions using
ruler and compass:
3. Given an angle find a bisector. (Now,
describe and explain your solution!).
In
this examples we start from using compass as an instrument for drawing circles,
and apply our understanding (coming from the construction) of circle as all points equidistant from the given point to
construct required points. Statements and their proofs come as byproduct of the
process of making sense and motion from what we can do to what is required. For
example, we can state that all three
perpendicular bisectors of sides of a triangle intersect at one point. The proof of the statement comes from the
construction we have.
Notes on
“Mathematics
curriculum development for computerized environment: a
designer-researcher-learner-teacher activity.”
In the
Geometry section the paper talks
about two important points:
and
Both
items can be given proper attention by the use of artifacts, including computer
with geometrical software. In this regard the authors underline the importance of task for students in
achieving both:
- flexibility in understanding of geometrical objects as
concepts (e.g. circle is not just a round but the set of points with certain property, which is critical in all
constructions) and
- seeing the necessity for proofs and learning to explain
and reason.
In
section on Algebra, they face the pedagogical
problem related to the possibility to describe multiples of 5 in two algebraic ways:
-
by explicit formula,
e.g. f(n) = 5*n and
-
by recurrence
relation, e.g. f(n+1) = f(n)+5, f(0) = 0,
Note
that as n changes n=0,1,2,3,.. function
f(n) has the same value in both cases:
|
N |
0 |
1 |
2 |
3 |
|
f(n) |
0 |
5 |
10 |
15 |
What
is different is the way of producing the
values in each case, the algorithmic, and thus, internal way of thinking about the function.
Students
used spreadsheets to experiment with
numbers in order to develop the idea of algebraic
expression from generalizing patterns.
Specific
(syntactic) features of the program led the students to improper written
expression (such as f(n)=n+5 in place of either of
expressions given above). This highlights the need for teacher’s attention to
correct the student in situations like that.
Similar situations constitute a potential danger for developing
inconsistent way to record and communicate mathematical expressions.
Note
that spreadsheets were not created to
support the development of mathematical language, but this software was chosen
based on the criteria of generality and accessibility as well as a tool for
mathematical computations and experiments with numbers.
Teachers’
awareness about a mathematical fact leading to possible learning complication
and errors gives an opportunity to discuss this situation with the class, and
thus convert a dangerous spot into a
learning benefit.
Notes on
The ideas and additional background:
1. Passing from oral to written language is an example of technological revolution.
“Writing creates the difference in not only the expression of thoughts but also how the thoughts are perceived.” The difference between oral and written languages is related to the deep difference between formal and informal mathematics. (We have seen a similar idea in Chap 29.) Interaction between the learner and computer is symbolic based and often requires interpretation of written symbols. This contains a potential of connecting oral and written strategies in mathematical problem solving.
2. “How do particular features of computer environment function cognitively?” is a complex question. The paper tries to partly focus on it discussing some principles of introduction of computers in school practice.
3. According to Dewey, act of thought has five steps: (1) perception of difficulty,
(2) determination of difficulty, (3)proposal of solution, (4) development of consequences of the solution, (5) further considerations leading to acceptance or dejection of the solution; In this respect the paper emphasizes that computers might change:
4. The authors build their approach on the following theoretical grounds: Constructivism, Instrumentalism, and Mediation, with references to the Theory of Didactical Situations and Conceptual Fields Theory.
5. Constructivism gives us an idea of how computers may increase accessibility of learned material. For that it is important to note that concreteness and abstractness are very subjective judgments which characterize the learner’s relation to an object rather than the object itself. Same topic may be very concrete for one student and very abstract to another one. That depends on her field of experiences and practice. Focusing on the interacting between knowledge and individuals, the paper suggests that the interplay between abstract and concrete is the source of meaning, and that it is desirable to produce an educational environment in which abstracting become lived-in culture of experience, allowing the user to act in the virtual reality (micro-world). Microworlds allow the learner to relate objects and actions.
6. Reflective abstraction is a process of concept formation as a
result of problem solving: concept emerges from problems to which it
provides a solution. Constructivism emphasizes the significance of this
process for learning.
7. In his book “Theory of didactical situations in mathematics”, Guy Brousseau distinguishes between Non-didactical, Didactical and A-didactical situations (with respect to knowledge S). Recall that the Non-didactical situation is implicitly organized to encourage knowledge S; the second one (Didactical) is explicitly designed to learn knowledge S; the third one (A-didactical) situation contains all conditions which permit the students to establish a relationship with S, regardless of the teacher. In the latest scenario the teacher provokes expected learning adaptations by her choice of problems. “The problems must make the students act, speak, think, and evolve their own motivation… The students know that the problem was chosen to help them to esquire new piece of knowledge, which are entirely justified by the internal logic of the situation, and students can construct it without appealing to didactical reasoning.” (Brousseau).
This echoes with the constructivist’s thesis: learning results from a process of active adaptation.
8. In his Conceptual Fields Theory, Vergnaud defines concept as a triple:
9. There is something to remember when applying the above theories:
10. Sometimes students reaction may not have the meaning that teacher expected because the same artifact may be used by different students as a different instruments.
Analysis of the utilization schemas becomes fundamental to design tasks and activities involving calculators. User of an artifact might not access the meaning incorporated in the artifact:
- counting in decimal system using abacus does not imply understanding of the concept of base.
- drawing round shape using compass does not imply the concept of circle as locus of points equidistant from the center.
But having the field of practice available, the user of an artifact is better prepared for the concept appearance.
11. Mediation function of a computer is related to the possibility to share common language of communication. According to Vygotskij, externally oriented mediation of a technological tool (abacus, compass, computer) and internally oriented mediation of a psychological tool (sign) are linked by the process on internalization. For example, notational system for writing numbers has its origin in abacus-artifact designed as a tool to keep track of counting. Compass as a tool leads to the mathematical notion of circle when its use can be substituted by corresponding mental image of the action and the result. Teachers’ role is to guide the evolution of the meaning. Students use the tool to complete the task, and meaning occurs from internalization. Meanings are rooted in the action but their evolution is achieved by means of social construction in the classroom under the guidance of the teacher.
12. One of the known difficulties in learning formal Euclidian proofs is to understand what is given (axioms) and what is to be proven (theorems). In this respect the richness on environment (Cabri) may be an obstacle to grasp the meaning of a proof because the statement can be “observed directly”. For that reason, the students were given an option to choose the axioms themselves and proceed from them. (p.715)
Still it was observed that students have difficulties to separate creation of the drawing and identifying geometrical relationships.
13. Internalization of the dragging function (supported by Cabri) helps to transform perceptual data (continuous family of drawings) into a conditional relationship (formal geometrical statements).
14. Introduction of computers in schools requires a radical change of objectives and activities. The potential of the introduction is rooted in the fact that computers are product of human culture and thus incorporate human knowledge and experiences. The role of teacher still remains central, complex and delicate; it supersedes that of selecting a good problem and analyzing the artifact and its potential for learning mathematics.
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