Reasoning as a mathematical habit of mind - m-a.org.uk · Types of mathematical reasoning Deductive...
Transcript of Reasoning as a mathematical habit of mind - m-a.org.uk · Types of mathematical reasoning Deductive...
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What is Teaching?
Creating a
common experience
to reflect on
and so bring about learning.
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True or false?
39 x 46 = 39 x 45 + 46
39 x 46 = 39 x 45 + 39
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Much mathematical
reasoning is
independent of
arithmetical fluency.
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True or false?
39 x 46 = 39 x 45 + 39
3 x 5 = 3 x 4 + 3
326 x 18 = 326 x 17 + 326
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Reasoning –What is it?
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Reasoning
Problemsolving
WHAT
WHY
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Additive reasoning
Multiplicative reasoning
Probabilistic reasoning
Statistical reasoning
Proportional reasoning
Geometric reasoning - positional
Geometric reasoning - axiomatic
Approximate reasoning
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Types of mathematical reasoning
Deductive• Draw conclusions based on known facts
Inductive• Generate examples, ‘sniff out’ patterns
Abductive• Conjecture as to why?
Analogical• Identify and draw on similarities with things already
understood
Relational• Identify connections between quantities or shapes that do
not require numbers or measures
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Deductive
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Magic squares
3 2 4
2 1 1 5 2 2
Sum of 9 Sum of 6 Sum of 8
Given three numbers in the these positions,
and the sum, when can you create a magic square?
Source: Arcavi (1994)
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Inductive
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High-low difference
Write down three digits
1. Arrange the digits from largest to smallest
2. Arrange the digits from smallest to largest
3. Find the difference between the two numbers.
Repeat with the answer
What do you notice?
What do you wonder?
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Abductive
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Potential to reason
5 x 6 = 30
7 x 8 = 56
4 x 5 = 20
6 x 7 = 42
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Potential to reason
5 x 6 = 30 4 x 5 = 20
7 x 8 = 56 5 x 6 = 30
4 x 5 = 20 6 x 7 = 42
6 x 7 = 42 7 x 8 = 56
What do you notice?
What do you wonder?
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Analogical
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Analogical
A mathematical answer to 27 divided by 6 is 4 remainder 3.
Make up a real world problem that involves 27 divided by 6 but where it makes sense to round the answer up to 5.
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Relational
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Tea party
On Saturday some friends came to tea.
We shared a packet of biscuits.
On Sunday I had another tea party.
More friends came round and we shared the same number of biscuits.
Did each person on Sunday eat more, less or the same as each did on Saturday?
Source: Susan Lamon (2005)
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BiscuitsFriends
Fewer Same More
Fewer
Same
More Less
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BiscuitsFriends
Fewer Same More
Fewer More More
Same Less Same More
More Less Less
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Mathematical reasoning
• In 7 out of 9 cases you can find the answer without knowing any quantities
• Relational reasoning is independent of arithmetical ability
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BiscuitsFriends
5 7 5 5
8 8 6 8
5 7 5 49 8 6 7
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Relational
Sue and Julie are cycling equally fast around a track.
Sue started first.
When Sue had cycled 9 laps, Julie had cycled 3 laps.
When Julie completed 15 laps, how many laps had Sue cycled?
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Types of mathematical reasoning
Deductive
Inductive
Abductive
Analogical
Relational
• Children, from an early age, engage with all these types of reasoning.
• They are not easy to separate.
• We can help children learn more about mathematical reasoning.
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DCSF-RR118
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Mathematical reasoning, even
more so than children’s
knowledge of arithmetic, is
important for children’s later
achievement in mathematics.
Nunes et al. DSFC RR-118
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Reasoning –Why is it difficult
to enact?
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National Curriculum Aim
The national curriculum for mathematics aims to ensure that all pupils:
• reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
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Myths
It’s a ‘big production number’
Fluency and problem solving have priority
It’s only for the ….
It’s ‘developmental’
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One of the major errors of twentieth-century psychology was to suppose that there are childish ways of knowing which are outgrown, and ought to be transcended, as one grows up. The childish ones are the bodily ones, and are to do with concrete action and experience. The grown-up ones are abstract, logical and propositional. But it is a Cartesian mistake to think that, once you have mastered logic you don’t need the body anymore. … We should think of the developing mind as a tree that grows new branches, not as a spaceship whose booster rockets fall away for ever one they have done their job and are spent.
Claxton, 2015, p 165-66
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How we reason
mathematically is does
not change substantially
over the years.
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What we reason about
mathematically can
change dramatically
over the years.
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Challenges
'Push back’
Planning
Teaching demands
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Maintaining mathematical challenge
Group of teachers planned tasks with researchers
Focus on reasoning and problem solving
68 lessons
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Maintaining mathematical challenge
When taught two thirds of activities ‘declined’ into
• Procedures
• Non-systematic
• Non mathematical
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Maintain high level
Source: Henningson & Stein (1997)
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Moving from Fast to Slow thinking …
… results in a slight, but noticeable,
feeling of depression
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Objects of learning
Indirect• Fluency• Problem solving • Reasoning
Direct• Fractions• Multiplication• 3-D shapes
CONTENT
AIMS
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Teaching demands
Requires making sense of learners’ methods
AND
aligning these with accepted mathematical knowledge.
TALK central to both of these
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Discussion
or
Dialogue?
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Discussion
David Bohm likened discussion to throwingopinions back and forth in an attempt to convince each other of the rightness of a particular position.
The whole view is becomes fragmented and shattered into many pieces.
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Dialogue
Dialogue requires willingness and skill to engage with minds, ideas and ways of thinking other than our own; it involves the ability to question, listen, reflect, reason, explain, speculate and explore ideas; to analyse problems, frame hypotheses, and develop solutions; …
Dialogue within the classroom lays the foundations not just of successful learning, but also of social cohesion, active citizenship and the good society.
Robin Alexander
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Dialogic talk
Speaking AND listening
Building on ideas, not reporting•Repeat
•Re-voice
•Rephrase
•Build on
•Comment on
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StructureFreedom
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Experienced teachers do two apparently contradictory things: They use more structures, and yet they improvise more. … The challenge facing every teacher and every school is to find the balance of creativity and structure that will optimisestudent learning. Sawyer
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Five orchestrating practices
Anticipating
Monitoring
Selecting
Sequencing
Making connections between student responses
(Source: Stein, Engle, Smith, Huges, 2009)
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Reasoning –A mathematical
habit of mind
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Mathematical power is best described by a set of habits of mind. People with mathematical power perform thought experiments; tinker with real and imagined machines; invent things; look for invariants (patterns); make reasonable conjectures; describe things both casually and formally (and play other language games); think about methods, strategies, algorithms, and processes; visualize things (even when the "things" are not inherently visual); seek to explain why things are as they seem them; and argue passionately about intellectual phenomena.
Cuoco, Goldenberg and Mark (1996)
Mathematical power
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Reasoning chains
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Addition and subtraction
5 + 5
5 + 6
10 + 10
10 + 9
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Addition and subtraction
62 + 10
62 + 30
62 + 29
54 + 19
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Addition and subtraction
62 – 10
62 – 30
62 – 29
54 – 19
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Multiplication and division
6 x 10
6 x 40
6 x 39
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Multiplication and division
21 ÷ 7
70 ÷ 7
91 ÷ 7
270 ÷ 27
297 ÷ 27
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Multiplication and division
160 ÷ 16
320 ÷ 16
320 ÷ 32
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True or false?
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Statements or questions
If you add four even numbers, the answer is a multiple of 4.
If you add four even numbers, is the answer always a multiple of 4?
Responding to statements is more productive of reasoning than responding to questions.
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True or false?
20 + 8 = 10 + 18
20 + 8 = 19 + 9
80 + 8 = 60 + 18
8 + 90 = 40 + 50 + 8
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True or false
37 + 56 – 56 = 37
458 + 347 – 347 = 458
458 + 347 – 347 = 347
ConjectureIf you add a number and then take the same number away, you have the number you started with.
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True or false?
56 – 38 = 56 – 37 – 1
56 – 38 = 56 – 39 + 1
56 – 38 = 56 – 36 – t
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True or false?
64 ÷ 14 = 32 ÷ 28
64 ÷ 14 = 32 ÷ 7
42 ÷ 16 = 84 ÷ 32
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True or false?
A semi circle is always half a circle
Half a circle is always a semi-circle
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ALWAYS …
… needs to be given a special role in mathematical dialogue
• Do you think that will always be true for …..?
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Open number sentences
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Open number sentences
64 + 14 = [ ] + 64
64 + [ ] = 18 + 64
64 + [ ] = 18 + 62
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Open number sentences
647 – 285 = [ ] – 300
671 – 285 = 640 – [ ]
[ ] – 285 = 640 – 285
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Open number sentences
3 x 5 = 5 x [ ]
3 x 5 = 3 x 4 + [ ]
3 x [ ] = 3 x 5 + 3
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Open number sentences
70 ÷ [ ] = 7
[ ] ÷ 2 = 35
[ ] x 3.5 = 7
3.5 x 20 = [ ]
70 ÷ [ ] = 3.5
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Fill in the blanks
___ is 6 more than ___
____ is ____ plus 9
7 is ____ minus _____
Create 3 different answers for each.
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Fill in the blanks
____ is 3/5 of ____
8 is _____ (fraction) of _____
_____ is _____ (fraction) of 24
Create 3 different answers for each.
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From
•Answers as the product
To
•Answers as part of the product
Encouraging reasoning requires a shift
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How do I help learners get the answer to this problem?
To
What mathematics are they expected to learn from working on this problem?
A shift from asking
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John Dewey – Business of education
“While it is not the business of education … to teach every possible item of information, it is its business to cultivate deep-seated and effective habits of discriminating tested beliefs from mere assertions, guesses, and opinions.”
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References
Arcavi, Abraham. (1994). Symbol sense: informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24–35.
Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for a mathematics curriculum. Journal of Mathematical Behavior, 15, 375–402.
Henningsen, M., & Stein, M. K. (1997). Mathematical Tasks and Student Cognition: Classroom-Based Factors That Support and Inhibit High-Level Mathematical Thinking and Reasoning. Journal for Research in Mathematics Education, 28(5), 524. https://doi.org/10.2307/749690
Lamon, S. (2005). Teaching Fractions and Ratios for Understanding. Essential Content Knowledge and Instructional Strategies for Teachers, 2nd Edition. New York and London: Routledge.
Sawyer, K. (Ed.). (2011). Structure and Improvisation in Creative Teaching. Oxford ; New York: Oxford University Press.
Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313–340. https://doi.org/10.1080/10986060802229675