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# Mathematical Understanding (BH,CM,EK,LU)

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## Ellie Kirby

on 29 April 2014

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#### Transcript of Mathematical Understanding (BH,CM,EK,LU)

(cc) photo by Metro Centric on Flickr
(cc) photo by Franco Folini on Flickr
(cc) photo by jimmyharris on Flickr
(cc) photo by Metro Centric on Flickr
Developing Mathematical Understanding through Problem Solving and Open-ended Investigations:
by Charlotte Maloney, Beth Hodgkinson, Ellie Kirby and Lucy Urwin
Using & Applying
Thank You!
SKEMP
Haylock & Cockburn
Teacher: What is half of 12?

Harry: 1 and 2

Teacher: Can you use cars to split 12 into 2 groups?

Harry: 6, I get it now. It is the opposite to doubling!

Harry - Aged 5
Two types of understanding:
(1989)
"Rules without reason..." (Skemp, 1989;2)
Dixon (2003) - Habit Learning
Instrumental Understanding:
Relational Understanding:
"Knowing what to do and why..." (Skemp, 1989;2)
Dixon (2003) - Intelligent Learning
Example of Instrumental Understanding
Example of Relational Understanding
Vinner
Tall

Thompson
Dreyfus
"Children are empty vessels waiting to be filled by the wisdom and knowledge of their elders..."
VS
Understanding
Image - Definition
CONSTRUCTIVIST
Approaches
(2013)
Concrete materials
Symbols
Language
Pictures
Transmission approach = Teacher transfers knowledge through a didactic approach.

Discovery approach = Learning based on discovery and exploration - no structured teaching.

Connectionist approach = Teacher enables children to connect experiences to methods and reasons.
(1997)
Building Connections
(1992)
(1989)
(1994)
(1990)
Concept Image, Concept Definition
“There is little point in pupils being ‘numerate’ if they cannot apply what they know” (Hughes
et al
, 2000;2)

“The ability to solve problems is at the heart of mathematics” (Cockcroft, 1982)

“A mathematics curriculum without problem solving can be likened to a diet of PE in which children practise football or netball skills but never get to play the game” (Thompson, 2010; 8)

What is Problem Solving?
3Approaches

“They need to understand the mathematics they learn so that they can be creative in solving problems, as well as being confident and fluent..” (Ofsted, 2012: 4)
National Curriculum (2013)
“Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas” (DfES, 2013)
Employment Equation (2013)
“Their ability… to play a productive role in the workforce will depend on their mathematical competence” (Hodgen and Marks, 2013: 3) Supported by Cockcroft (1982)

Links to Pricthard's Extrinsic motivation. (Rogers, 1986)
Links to Pricthard's Intrinsic motivation. (Rogers, 1986)
Cognitive connections
Based on existing schema
Mathematical concept
Children already have experience with 'take-away' before entering school.
Children taught the mathematical definition (concept) of subtraction and are then able to complete a subtraction question.

Open
Problems
Brown Rhodes Johnson Wiliam
Charles Dickens as Cited in Hughes (2009;6)
CONCEPTUAL
&

PROCEDURAL
(Russell, 2000)
Learning with steps and procedures.
Rich connections - Allows children to reason and apply knowledge accurately.
THEORIES AND IDEAS
Dubinsky
Action
Process
Object
Schema
This links to Hewitt's (1999) Arbitrary and necessary understanding.
"A simple model that enables us to talk about understanding in mathematics is to view the growth of understanding as the building up of cognitive connections"
(Haylock and Cockburn, 2013; 9)
Sfard
Liebeck
Dubinsky

Liebeck
Sfard
(1991)
(1991)
Structural - as objects

Operational - as processes

"Sequence of Abstraction"
Experience Language Pictures Symbols (Liebeck, 1984)
(1984)
Teachers should encourage children to create mental images in their head.
Skemp (1989) Haylock (2006) Haylock and Cockburn (2008) (2013) Askew et al (1997) Vinner (1992) Sfard (1991) Liebeck (1984) Dubinsky (1991)Tall (1989) Thompson (1994) Dreyfus (1990) Cottrill (2003)
The term understanding has many different dimensions.
"The concept of mathematical understanding is central to curriculum development, classroom interaction, and the training of mathematics teachers" (Mousley, 2006; 553)
Atkinson
(1992)
"Any individual teacher may show some characteristics from each of these three models..." (Askew, 1997:3)
"The ability to apply mathematics to a variety of situations." (Cockcroft, 1982:73)
"Flexible, adaptable, multi-skilled problem solvers." (Pollard, 2005:275)
Closed
Open-ended
Word problems
Investigations
"Parts of a connected whole"
(Skemp, 1989;11)
(1991)
(1991)
(1984)
As cited in Cottrill (2003)
Ofsted (2012)
Ofsted (2008)
Cockcroft (1982)
National Curriculum (2013)
Problem Solving
Connectionist theories:
Skemp (1989)
et al
(1997)
Haylock & Cockburn (2013)
Liebeck (1984)
Dubinsky (1991)
Sfard (1991)
Vinner (1992)
Tall (1989)
Thompson (1994)
Dreyfus (1990)
Open
Investigations
The route to the solution is not immediately obvious. (Thompson,2003; Haylock, 2006; Haylock and Cockburn, 2013)
(Thompson, 2003; Haylock and Cockburn, 2013; Way, 2013)
Clear goal
Less specific goal
Multiple routes
Prescribed direction of enquiry
Children determine own direction of enquiry
Explore patterns
Benefits
Issues
Hard to manage on a whole class level
Teacher subject knowledge
Less control
Teaching to the test
Time constraints
Lack of written evidence
Higher order thinking skills
Motivational tool
Whole school approach
Promote mathematical discussion
Cater for a wide range of needs
Valuable tool of assessment
Lack of
Subject Knowledge
& Expertise
The quality and consistency of teaching varies too much. (Ofsted, 2012)

Investigations may leave some teachers feeling less 'in control' (Thompson, 2003)
Higher Order Thinking Skills
Reasoning
'I know that 3's go into 36, so there's 1 left over'
Information
Processing
'I can sort these into groups of 3 and 5'
Creative
Thinking
'How many different ways can we...?'
Enquiry
'What are we trying to find out?'
Communication
'I tried it like this first'
Evaluation
'It might have been easier if we sorted them into groups'
(Koshy and Murray, 2011; Thompson, 2003; O'Sullivan
et al
, 2005)
(See page 1)
Motivation
Teaching to the Test
& Time Constraints
Mathematical Discussion
Assessment
Benefits
Drawbacks
Cater for all Abilities
Investigations "must to be clear and meaningful and make 'human sense' to the child." (Atkinson, 1992;27)
(Ofsted, 2008; Ofsted, 2010; Ofsted, 2012; Skemp, 1989; Thompson, 2003)
"The backwash effect of examinations" (Skemp, 1989;11)
"Over- burdened syllabi" (Skemp, 1989;11)
(Black and William , 2001; Clarke, 2001; Clarke, 2005)
1. Feedback
2. Active involvement of children in their learning
4. Motivation and self-esteem
5. Self assessment
Principles of Formative Assessment
(Way, 2013)
hidden talents
Collaborative
Group Work
Inter-thinking is "our use of language for thinking together, for collaboratively making sense of experience and solving problems."
(Mercer, 2000;1)
Group discussion gives children "the opportunity to participate more directly, share their ideas and extend their learning within a small group of peers."
(Williams, 2008;67)
(See page 3)
HA
MA
LA
Extend
Support
Way, 2013; Vinner, 1992; Pound and Lee, 2011; Haylock and Cockburn, 2013
SOCIAL-CONSTRUCTIVIST
CONSTRUCTIVIST
Bruner - Intrinsic satisfaction

Piaget - Stages of cognitive development

Vygotsky - Scaffolding
Liebeck
(1984)
Whole school approach can further increase motivation.
(Atkinson, 1992; Ofsted, 2012)
(Koshy & Murray, 2011)
(As cited in Cottrill, 2003)
(Cited in Cottrill, 2003)
(Cited in Cottrill, 2003)
(Cited in Cottrill, 2003)
(Cited in Cottrill, 2003)
(Cited in Cottrill, 2003)
(Cited in Hughes, 2009)
(Cockcroft, 1984; Atkinson, 1992; Carruthers and Worthington, 2006)
(Thompson, 2003)
(See page 2)
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