reasoning and connection across a-level mathematical concepts

REASONING AND CONNECTION ACROSS A-LEVEL MATHEMATICAL

CONCEPTS

Dr Toh Tin LamNational Institute of Education

COMMUNICATION, REASONING & CONNECTION

• Singapore Mathematics Framework• Reasoning• Connection• Communication

Singapore Mathematics Framework

NumericalAlgebraicGeometricalStatisticalProbabilisticAnalytical

Monitoring of one’s own thinkingSelf-regulation of learning

Beliefs

Interest

Appreciation

Confidence

Perseverance

Concepts

Processes

Attitudes

Metacognition

Skill

sMathematical

Problem Solving

Numerical calculationAlgebraic manipulation

Spatial visualisationData analysisMeasurement

Use of mathematical toolsEstimation

Reasoning, communication and connectionsThinking skills and heuristicsApplications and modelling

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

)(sin x

dx

d

xxdx

dcos)(sin

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?

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.

,3 ,1

1

nuu

n

iin

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)

1

0

21 dxx 5.0

0

21 dxx

3

3

2011sin xdx

CONNECTION

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

Light ray

2

1

1

1

2

1

r

3

1

2

1

rPlane

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.

22 )1(

1

x

22 )1(

1

x

CONNECTION

• To connect a solution to real world situation..

hkdt

dh

Leaking Bucket:

CONNECTION

hkdt

dh

Leaking Bucket:

Solving the differential equation,

Does it make sense?

20 )2(

4

1kthh

CONNECTION

20 )2(

4

1kthh

k

ht

k

htkth

h0

020

2,0

2,)2(

4

1

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

610C !4!6

!10

rnrr

n ppCrXP )1()(

COMMUNICATION

• Are the following statements TRUE?

baba

11

22 baba

cbcaba

cbcaba

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