Mathematical generalization

Mathematical generalization is topic covered under pedagogical content/ methods of teaching mathematics in B.Ed/ B.Sc.B.Ed integrated courses.

1. METHODS OF TEACHING  MATHEMATICS

2. TOPIC: MATHEMATICAL GENERALIZATION Presented by: GUNJAN AWASTHI                       B.Sc.B.Ed . 5th sem.                     Roll no. 13

3. A lesson without the opportunity for learners to generalise is not a mathematics lesson.                                            - J Mason, 1996

4.        MATHEMATICAL  GENERALIZATION   In mathematics, generalization can be both a process and a product.  When one looks at specific instances, notices a pattern, and uses inductive reasoning to conjecture a statement about all such patterns, one is generalizing. The symbolic, verbal, or visual representation of the pattern in your conjecture might be called a generalization.

5. What is generalization? There are three meanings attached to generalization from the literature. The first is as a synonym for abstraction. That is, the process of generalization is the process of “finding and singling out [of properties] in a whole class of similar objects. In this sense it is a synonym for abstraction. The second meaning includes extension (empirical or mathematical) of existing concept  or a mathematical invention. The third meaning defines generalization in terms of its product. If the product of abstraction is a concept, the product of generalization is a statement relating the concepts, that is, a theorem.

6. By defining examples such as: a2 × a3 = (a × a) × (a × a × a) = a5 a3 × a4 = (a × a × a) × (a × a × a × a) = a7 and so on One can conclude that: am × an = am+n …thus generalizing to all cases for a specific domain for the base “a” and the exponents “m” and “n.”

7. One more example- When a student notices that the sum of an even and an odd integer always results in an odd integer, that student is generalizing.   Generalizations such as this allow students to think about computations independently of the particular numbers that are used. Without this, and many other generalizations made in mathematics from the early grades, all work in mathematics would be cumbersome and inefficient.

8. Role of generalization in advanced mathematical thinking Generalization and abstraction both play an important role in the minds of mathematics students as they study higher-level concepts.generalization as the derivation or induction from something particular to something general by looking at the common things and expanding their domains of validity. As we teach our own math courses, we can look out for opportunities to introduce generalization and abstraction in order to help our students better understand the pattern behind what they are learning.

9. Importance of mathematical generalization in teaching mathematics There are several advantages to applying generalization in our math classes, and its positive effect on teaching and learning is a fundamental way to provide our students with the tools needed for successful advanced thinking in mathematics .

10. Generalizing is the process of "seeing through the particular" by not dwelling in the particularities but rather stressing relationships… whenever we stress some features we consequently ignore others, and this is how generalizing comes about.                                                                                         - Mason

11. Thank you