Noel-Ann Bradshaw

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Encouraging Maths Students to Think

• Noel-Ann Bradshaw
• University of Greenwich

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Outline

• Can maths students think?
• What we have done to encourage this?
• How students have reacted to this?
• What else we can do?
• Discussion

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Motivation

• Thinking Mathematically by John Mason
• What to do when stuck
• Specialisation
• Generalisation
• Improved grades!
• Mathematical Technology and Thinking

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Specialisation
Differentiate the following w.r.t.
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Specialise then Generalise

• A number like 12321 is called a palindrome
  because it reads the same backwards as forwards.
  A friend of mine claims that all palindromes with
  four digits are exactly divisible by eleven. Are
  they?

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General form

• A four-digit palindrome is of the form
• abba
• 1000a 100b10ba (10001)a (10010)b
• 1001a
  110b
• 
  11(91a10b)
• Proved with a general form

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How does it feel to be stuck?

• What does it feel like to be faced with a problem
  you cant answer?
• What does it feel like to be stuck?
• How can you begin to specialise?
• How can you turn this into generalising?
• Do you always have to do this?
• Does it help why?
• Record your feelings

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What do you know and what do you want to know?

• Our three ages combined amount to just seventy
  years. As I am just six times as old as you are
  now, it may be said that when I am but twice as
  old as you, our three ages combined will be twice
  what they are at present. Now let me see if you
  can tell me how old your mother is?

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Response to being stuck

• Stare at a blank page
• Do anything except the problem you are trying to
  do
• Get stressed / panic
• Feel frustrated / useless etc
• Do what has been suggested, take a break and try
  again

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Write down your thoughts

• What numbers have an odd number of divisors?

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Consecutive sums

• Some numbers can be expressed as the sum of a
  string of consecutive positive numbers.
• Exactly which numbers have this property?
• Eg 9234
• 11 56

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Maths Café Question

• 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
  15, 16
• Can you separate the integers 1 to 16 into two
  sets of equal sizes so that each set has the same
  sum, the same sum of squares, and the same sum of
  cubes? If yes list the two sets!

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Other tools

• Maths busking
• Visual demos

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Student feedback

• Not entirely positive!
• Introduce more across all four first year courses
• Build on this in the second year
• See the point in final year

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Discussion

• Do you find that your students are resistant to
  thinking?
• If so, how do you encourage them to think?