Reasoning and connection across a level mathematical concepts
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REASONING AND CONNECTION ACROSS A-LEVEL MATHEMATICAL CONCEPTS. Dr Toh Tin Lam National Institute of Education. COMMUNICATION, REASONING & CONNECTION. Singapore Mathematics Framework Reasoning Connection Communication. Singapore Mathematics Framework. Beliefs Interest Appreciation

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REASONING AND CONNECTION ACROSS A-LEVEL MATHEMATICAL CONCEPTS

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REASONING AND CONNECTION ACROSS A-LEVEL MATHEMATICAL CONCEPTS

Dr Toh Tin Lam

National Institute of Education


COMMUNICATION, REASONING & CONNECTION

  • Singapore Mathematics Framework

  • Reasoning

  • Connection

  • Communication


Singapore Mathematics Framework

Beliefs

Interest

Appreciation

Confidence

Perseverance

Monitoring of one’s own thinking

Self-regulation of learning

Metacognition

Attitudes

Mathematical

Problem

Solving

Reasoning, communication and connections

Thinking skills and heuristics

Applications and modelling

Numerical calculation

Algebraic manipulation

Spatial visualisation

Data analysis

Measurement

Use of mathematical tools

Estimation

Processes

Skills

Concepts

Numerical

Algebraic

Geometrical

Statistical

Probabilistic

Analytical


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)


REASONING

  • Typical Class ...


REASONING

  • Typical Class ...


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


REASONING

  • Given a new problem, a problem situation imageis structured. Tentative solution starts arise from the problem situation image.

    (Selden, Selden, Hawk & Mason, 1999)

  • How should the tentative solution starts be anchored?


REASONING

  • Would you want to infuse some reasoning into this chapter?


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!


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...


REASONINGS

  • Cambridge exam question (J87/S/1(b))

    The sequence u1,u2, ...... , un ,...is defined by

    and u 1 =1, u2 = 1. Express un in terms of n and justify your answer.


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).


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)


CONNECTION

  • Difficulties of students making connections across different concepts....


CONNECTION

  • Involve students in more opportunities to connect different concepts:

    Evaluate (a)(b)

    (c)


CONNECTION

  • In greater ways..... Have a “big” question that summarizes a big chapter.

Light ray

Plane


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.


CONNECTION

  • To connect a solution to real world situation..

Leaking Bucket:


CONNECTION

Leaking Bucket:

Solving the differential equation,

Does it make sense?


CONNECTION


CONNECTION

  • 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


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


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.


SUMMARY

REASONING

COMMUNICATION

CONNECTION


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