1. REASONING AND CONNECTION ACROSS A-LEVEL MATHEMATICAL CONCEPTS Dr Toh Tin Lam National Institute of Education

2. COMMUNICATION, REASONING & CONNECTION Singapore Mathematics Framework Reasoning Connection Communication

3. Singapore Mathematics Framework Numerical Algebraic Geometrical Statistical Probabilistic Analytical Monitoring of one’s own thinking Self-regulation of learning Beliefs Interest Appreciation Confidence Perseverance Concepts Processes Attitudes Metacognition Skills Mathematical Problem Solving Numerical calculation Algebraic manipulation Spatial visualisation Data analysis Measurement Use of mathematical tools Estimation Reasoning, communication and connections Thinking skills and heuristics Applications and modelling

4. REASONING Mathematics should make sense to students. Students should develop an appreciation of mathematical justification in the study of all mathematical content. Students should develop a repertoire of increasingly sophisticated methods of reasoning and proofs. (NCTM, 2000)

5. REASONING Typical Class...

6. REASONING Typical Class...

7. REASONING Why does the rule hold true only when x is in radian? What happens with x is in degrees? What will the formula be? Can you follow through the first principle and give me the formula for

8. REASONING Given a new problem, a problem situation image is structured. Tentative solution starts arise from the problem situation image. (Selden, Selden, Hawk & Mason, 1999) How should the tentative solution starts be anchored?

9. REASONING Would you want to infuse some reasoning into this chapter?

10. REASONING What are the reasoning you would expect to see in this chapter (our e.g. Differentiation)? Even rule-based topics should be used to engage students in reasoning!

11. REASONINGS What type of reasoning & proofs would you like to see in JC mathematics classes? Pattern Gazing & Making Conjectures; Rigorous mathematical proofs  to build on making gazing and making conjectures...  deeper understanding of the proof itself...

12. REASONINGS Cambridge exam question (J87/S/1(b)) The sequence u 1, u 2,......, u n,...is defined by and u 1 =1, u 2 = 1. Express u n in terms of n and justify your answer.

13. REASONINGS What is wrong with the proofs? (Pg 1 & 2) Get students to critically assess the accuracy of the mathematical argument (deep thinking over the mathematical steps).

14. CONNECTIONS Learning of new concepts builds on students’ previous understanding Links across different topics of mathematics Ability to link mathematics with other academic disciplines gives them greater mathematical power (NCTM, 2000)

15. CONNECTION Difficulties of students making connections across different concepts....

16. CONNECTION Involve students in more opportunities to connect different concepts: Evaluate (a)(b) (c)

17. CONNECTION In greater ways..... Have a “big” question that summarizes a big chapter. Light ray Plane

18. CONNECTION Ways to link the different topics together. Small ways... (J88/S/Q1(b)) By considering the expansion of or otherwise, evaluate the n derivative of when x = 0.

19. CONNECTION To connect a solution to real world situation.. Leaking Bucket:

20. CONNECTION Leaking Bucket: Solving the differential equation, Does it make sense?

21. CONNECTION

22. An obvious disconnection.... Find the number of ways to permute 6 “s”s and 4 “f”s in a row. Is the answer or If X  Bin (n, p), then

23. COMMUNICATION Are the following statements TRUE? If you suspect a statement is TRUE, try to prove it; if you think that it is FALSE, try to look for a counter- example to disprove the statement. Get students to think over the logical statement. Lead students to communicate in acceptable mathematical language

24. COMMUNICATION Teachers: engage students in thought- provoking activities rather than simply telling them the method of solving a particular mathematics problem. Give students opportunity to explain their solution. Give students questions that require their explanation.

25. SUMMARY