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Charlie Gilderdale
University of Cambridge
December 2014
Mathematics Workshop 1:Developing active learners
Inspiring teaching,inspiring learning
Mathematics is a creative discipline, not a spectator sport
Exploring → Noticing Patterns
→ Conjecturing
→ Generalising
→ Explaining
→ Justifying
→ Proving
Some ways to make mathematical tasks more engaging
• reverse the questions
• look at/for alternative methods
• seek all possibilities
• greater generality (what if…?)
M, M and M
Can you find five positive whole numbers that satisfy the following properties:
Mean = 4
Mode = 3
Median = 3
Can you find all the different sets of five positive whole numbers that satisfy these conditions?
Why might a teacher choose to use this activity?
Some ways to make mathematical tasks more engaging
• reverse the questions
• look at/for alternative methods
• seek all possibilities
• greater generality (what if…?)
Cryptarithms
Two and Two
Extension:
Can you find other word sums that work? Here are some suggestions to start you off:
ONE + ONE = TWOONE + TWO = THREEONE + THREE = FOURFOUR + FIVE = NINE
Can you make a word subtraction?
Forwards add Backwards
726 can be formed by adding a 3-digit number with its reversal.
Can you find any other ways of making 726 in this way?
How about 707 and 766?
Which other numbers between 700 and 800 can be formed from a number plus its reversal?
Why might a teacher choose to use these activities?
Some ways to make mathematical tasks more engaging
• reverse the questions
• look at/for alternative methods
• seek all possibilities
• greater generality (what if…?)
Wipeout
One of the numbers from 1 2 3 4 5 6 is wiped out. The mean of what is left is 3.6Which number was crossed out?
One of the numbers from 1 2 3 4 5 6 7 is wiped out. The mean of what is left is 4.0Which number was crossed out?…
One of the numbers from 1 to N, where N is an unknown even number, is wiped out. The mean of what is left is an integer (whole number).Which numbers could have been crossed out?Can you explain why?
Why might a teacher choose to use this activity?
Some underlying principles
Mathematical tasks should address both content and process skills.
Rich tasks can replace routine textbook tasks, they are not just an add-on for students who finish first.
What Teachers Can Do
• Aim to be mathematical with and in front of learners
• Aim to do for learners only what they cannot yet do for themselves
• Focus on provoking learners to
use and develop their (mathematical) powers
make mathematically significant choices
John Mason
If I ran a school, I’d give all the average grades to the ones who gave me all the right answers, for being good parrots. I’d give the top grades to those who made lots of mistakes and told me about them and then told me what they had learned from them.
Buckminster Fuller, Inventor
Valuing mathematical thinking
Guy Claxton’s Four Rs
Resilience: being able to stick with difficulty and cope with
feelings such as fear and frustration
Resourcefulness: having a variety of learning strategies
and knowing when to use them
Reflection: being willing and able to become more
strategic about learning. Getting to know our own
strengths and weaknesses
Reciprocity: being willing and able to learn alone and with
others
Think of a topic you’ve just taught,or are about to teach,
and look for opportunities to
• reverse the questions
• look at/for alternative methods
• seek all possibilities
• greater generality (what if…?)
Learn more!Getting in touch with Cambridge is easy
Email us at
or telephone +44 (0) 1223 553554
www.cie.org.uk
