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REASONING AND CONNECTION ACROSS A-LEVEL MATHEMATICAL
CONCEPTS
Dr Toh Tin LamNational Institute of Education
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COMMUNICATION, REASONING & CONNECTION
• Singapore Mathematics Framework• Reasoning• Connection• Communication
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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
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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)
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REASONING
• Typical Class ...
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REASONING
• Typical Class ...
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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
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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?
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REASONING
• Would you want to infuse some reasoning into this chapter?
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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!
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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...
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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
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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).
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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)
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CONNECTION
• Difficulties of students making connections across different concepts....
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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
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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
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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
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CONNECTION
• To connect a solution to real world situation..
hkdt
dh
Leaking Bucket:
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CONNECTION
hkdt
dh
Leaking Bucket:
Solving the differential equation,
Does it make sense?
20 )2(
4
1kthh
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CONNECTION
20 )2(
4
1kthh
k
ht
k
htkth
h0
020
2,0
2,)2(
4
1
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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()(
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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
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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.
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SUMMARY