Mathematical process

Mathematical process , Reasoning , Examples,Types of Reasoning, Argumentation, Examples How to argument, Justification and Advantages , How to justify equation.

1. Government college of teacher education Chitradurga , 577501

2. Mathematical process. .... Mathematical process means thinking and reasoning, calculation and salvation of mathematical problems by mathematical methods. Ex:- Derive (a-b) ^3

3. (a-b) ^3= (a-b)(a-b)(a-b)} Thinking. (a-b)^3 = (a^2-2ab+b^2) (a-b) [Since, (a-b)^2 = a^2+b^2-2ab)} ( Reasoning) . (a-b)^3 = a^3-2a^2b+ab^2-a^2b+2ab^2-b^3
(a-b)^3 = a^3-3a^2b+3ab^2– b^3 (Calculation) Therefore, the formula for (a-b)^3 is a^3-3a^2b+3ab^2– b^3.

The above formula can also be written as:

(a-b)^3 = a^3-3ab(a-b) – b^3}( Solution) .

4. Mathematical process includes 1) Reasoning. 2) Logical thinking. 3) Problem solving. 4) Connecting. 5) Problem posing. 6) Abstraction. 7) Argumentation. 8) Justification. Etc

5. Mathematical processEs .

6. Mathematical process. 1) Mathematical reasoning. 2) Mathematical argumentation. 3) Mathematical justification.

7. Mathematical reasoning . Mathematical reasoning happens through making conjectures, investigating, and representing and finding and explains and justifying conclusions. Reasoning can be thought of as the process of drawing conclusions on the basis of evidence or stated assumption and sence making can be defined as developing and understanding of a situation, context, or concept by connecting it with existing knowledge.

8. Mathematical reasoning. The (NCTM) National council of teacher of mathematics defines reasoning as the “A productive way of thinking that becomes common in the process of mathematical inquiry and sence making”. Ex:- Solve (20)^2

9. Student 1 Solve (20)^2 Student 2 Solve (20)^2 Student 3 Solve (20) ^2 20×20 =400 (20)^2=(15+5)^2 (a+ b)^2 = a^2 + 2ab + b^2 (15+5) ^2=(15) ^2+2×15×5+(5) ^2 (20)^2=(25-5)^2 (a– b)^2 = a^2 – 2ab + b^2 (25-5)^2=(25)^2-2×25×5+(5)^2 (20)^2=400. =225+150+25 =400 (20) ^2=400. =625-250+25 =400 (20)^2=400.

10. Example of Reasoning. Ex:- Solve 18+27 =__+ 29 Here twenty nine is two more than 27 , So the number which is added has to be two less than 18 to making the eqution . Instead of adding 18+27 then figuring out the number to add 29 to get 45,simplify the calculation by comparing the numbers on both sides one who realised 29 is two more than 27.So the number added had to be two less than 18 that is 16.

11. Mathematical Reasoning Two types . 1) Inductive reasoning. 2) Deductive reasoning. Inductive reasoning:- Inductive reasoning is a logical process in which multiple premises, all believed true or found true most of the time, are combined to obtain a specific conclusion. Inductive reasoning is a method of logical thinking that combines observations with experiential information to reach a conclusion.

12. Inductive reasoning.

13. Inductive Reasoning.

14. Inductive Reasoning.

15. Deductive reasoning . Deductive reasoning is a logical process in which a conclusion is based on the concordance of multiple premises that are generally assumed to be true. Example All numbers ending in 0 or 5 are divisible by 5. The number 35 ends with a 5. so it must be divisible by 5.

16. Deductive Reasoning .

17. Deductive reasoning.

18. Difference between inductive and Deductive reasoning

19. Advantages of mathematical reasoning. Helps children to think logically and make sense of mathematics. Help children to test hypothesis. Help chldren to make predictions. Helps children to explain their thinking.

20. Mathematical argumentation.

21. Mathematical argumentation. A mathematical argumentation is sequence of statements and reasons given with the aim of demonstrating that a claim is true or false . OR The act or process of forming reasons and of drawing conclusions and applying them to a case in discussion

22. Mathematical argumentation example Statement 1:- Numbers ending with 0 and 5 are divisable by 5) How? Statement 2 :- Numbers ending with 0 divisable by 10.} How? Statement 3:- Numbers ending with 3 and 9 divisable by 2] How? Statement 1:- True Statement 2:- True Statement 3:- False

23. Mathematical Argumentation. Example

24. Mathematical argumentations are like this. How does that look like? What happened after? Can you tell me why? How does this fit?

25. Mathematical argumentation are like this. Can you explain? What happened before? What would happen if you used this number? What would change if..... Show me where..... What could you add to strengthen this part? How would that work?

26. Advantages of mathematical argumentation. Discover new mathematical ideas. Argumentation supports for the conclusion. Contributed to the class understanding. Tool for student reasoning. Provides evidence and reasoning for new idea.

27. Mathematical justification . Mathematical justification in a mathematical setting teaches a important writing style, writing a brief, information packed statement that gives the reader a solid reason to believe your conclusion, without wasting a readers time. OR Mathematical justification is the use appropriate mathematical language to give reasons for the particular approach used to solve a problem.

28. Example Determine when the first equation is cheaper than the second
C = 40t+200 C=30t+300
Developing a solution
Students will work on the solution of the problem. Either graphically, or by solving simultaneous equations, the time for which costs are equal is 10 hours.

Note: The time for which the first rate is more than the second is any time greater than 10 hours.

29. Mathematical justification properties.

30. How to justify equation

31. Advantages of mathematical justification. Students learn better when they self justify and and explain the solution. It supports logical reasoning and analytical thinking. It improves problem solving skills. Teachers need to know how students arrive at their answer. Explanations encourage students to explain the why and not just the how.

32. Thank you ....