How, Why and What of Algebra

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How, Why and What of Algebra

• Kaye Stacey
• University of Melbourne

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AIMS

• Continue international survey (Australia, Italy,
  Norway, USA, UK)
• Give further exemplification
• Think about putting research into a form that can
  affect practice

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Bushwalking with Kim. Kim wants to walk from A
to B through varying terrain. What route should
she take?
B
C
A
Faster here
Slower here
P
Q
R
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Yr 12 assessment version 2 km/hr in bush, 5
km/hr in clearing (45 deg) AB 14 km, CR
7 km, C midpoint
B
C
A
Faster here
Slower here
P
Q
R
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Formulation of the mathematical problem

• Total time
• sum of times on three segments
• distance 1 /speed 1
• distance 2 /speed 2
• distance 3 /speed 3
• AIM minimise total time

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Setting up the variables and expressions.
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• AP found by cosine rule
• d(x) long expression in x
• Define function
• t(x, b, c) 2x/c 2d(x)/b
• d(x) sqrt (2x2 72 -2.sqrt2.7.cos 45)

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Time calculated from raw distances
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Minimum time for b2, c5
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..or find a minimum graphically
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With CAS, can find minimum as function of k c/b

• and then explore what this answer means
• what if k1?, domain of validity of solution
• what if distances change?

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Bushwalking with Kim. Kim wants to walk from A
to B through varying terrain. What route should
she take?

• Solution 1 Algebraic formulation, with calculus
  for optimisation
• Solution 2 Algebraic formulation, with graphical
  optimisation
• Solution 3 Dynamic geometry formulation, with
  dragging optimisation

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Orienteering with Kim a numerical drag
solution can deal with a clearing in any position
Faster here
Slower here
B
A
Time dist 1 /speed 1 dist 2 /speed 2 dist
3 /speed 3
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Some observations . . .

• Our move to reduce symbolic manipulation has been
  good - but we need to develop algebraic
  expectation and symbol sense with only minimal
  practice.
• Emphasis on formulation and interpretation
  essential we still need more in schools (and
  more research).
• What would an intellectually strong numerical
  algebra be like? Can enough cognitive obstacles
  be avoided to justify big change ? (Note need
  for reification in both)
• Basic algebra can no longer be justified by
  pragmatic reasons - so we need to clarify the
  epistemic reasons.

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• A black box is OK, but a doubly black box is
  not!

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Thank you