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

REASONING AND CONNECTION ACROSS A-LEVEL MATHEMATICAL CONCEPTS

Dr Toh Tin Lam

National Institute of Education


Communication reasoning connection
COMMUNICATION, REASONING & CONNECTION CONCEPTS

  • Singapore Mathematics Framework

  • Reasoning

  • Connection

  • Communication


Singapore mathematics framework
Singapore Mathematics Framework CONCEPTS

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
REASONING CONCEPTS

  • 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)


Reasoning1
REASONING CONCEPTS

  • Typical Class ...


Reasoning2
REASONING CONCEPTS

  • Typical Class ...


Reasoning3
REASONING CONCEPTS

  • 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


Reasoning4
REASONING CONCEPTS

  • 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?


Reasoning5
REASONING CONCEPTS

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


Reasoning6
REASONING CONCEPTS

  • 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
REASONINGS CONCEPTS

  • 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...


Reasonings1
REASONINGS CONCEPTS

  • 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.


Reasonings2
REASONINGS CONCEPTS

  • 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
CONNECTIONS CONCEPTS

  • 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
CONNECTION CONCEPTS

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


Connection1
CONNECTION CONCEPTS

  • Involve students in more opportunities to connect different concepts:

    Evaluate (a) (b)

    (c)


Connection2
CONNECTION CONCEPTS

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

Light ray

Plane


Connection3
CONNECTION CONCEPTS

  • 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.


Connection4
CONNECTION CONCEPTS

  • To connect a solution to real world situation..

Leaking Bucket:


Connection5
CONNECTION CONCEPTS

Leaking Bucket:

Solving the differential equation,

Does it make sense?


Connection6
CONNECTION CONCEPTS


Connection7
CONNECTION CONCEPTS

  • 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
COMMUNICATION CONCEPTS

  • 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


Communication1
COMMUNICATION CONCEPTS

  • 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
SUMMARY CONCEPTS

REASONING

COMMUNICATION

CONNECTION


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