1 Teaching as The Directing of Attention and Learning as the Transformation of Attention 2007 John Mason Stowarzyszenia Nauczycieli Matematyki Muszyna.

Slides:



Advertisements
Similar presentations
1 The Role of Attention in Teaching & Learning Mathematics John Mason Stockholm Biennalen 2008.
Advertisements

Principles of Good Maths Teaching
$200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 Order of Operations Mean,
1 Noticing: the key to teaching, learning and research John Mason Modena & Napoli 2007.
1 When and How is Mathematics Actually Learned? John Mason Trondheim Oct 2007.
1 Asking Questions in order to promote Mathematical Reasoning John Mason East London June 2010 The Open University Maths Dept University of Oxford Dept.
1 Learning to Think and to Reason Algebraically and the Structure of Attention 2007 John Mason SMC.
1 Where is the Reality of Algebra & Geometry ? John Mason Surrey HoDs Feb 2009 The Open University Maths Dept University of Oxford Dept of Education.
1 Learning to Think and Reason Geometrically: The roles of monologue, dialogue, and multilogue and the Structure of Attention 2007 John Mason SMC.
1 Getting Students to Take Initiative when Learning & Doing Mathematics John Mason Oslo Jan 2009 The Open University Maths Dept University of Oxford Dept.
1 Developing Mathematical Thinking John Mason Flötur, Selfoss Sept 2008.
1 Structure of Attention UBC Vancouver July 2009.
1 Learner Generated Examples in the Teaching of Mathematics John Mason Grahamstown May 2009 The Open University Maths Dept University of Oxford Dept of.
1 Phenomenal Mathematics Phenomenal Mathematics John Mason AAMT-MERGA Alice Springs July The Open University Maths Dept University of Oxford Dept.
Generalisation in Mathematics: who generalises what, when, how and why? John Mason Trondheim April 2009.
1 Making the Most of Mathematical Tasks John Mason Overton Jan 2011 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical.
1 Attention to Attention in the Teaching and Learning of Mathematics John Mason Open University & University of Oxford Flötur Selfoss Sept 2008.
1 Necessary Movements of Attention John Mason ATM March 2009.
1 Fundamental Constructs Underpinning Pedagogic Actions in Mathematics Classrooms John Mason March 2009 The Open University Maths Dept University of Oxford.
1 Mathematics: with good reason John Mason Exeter April 2010 The Open University Maths Dept University of Oxford Dept of Education.
1 Wherein lies the Richness of Mathematical Tasks? John Mason Windsor & Datchett Feb 2008.
1 With and Across the Grain: making use of learners’ powers to detect and express generality London Mathematics Centre June 2006.
1 Progress in Mathematical Thinking John Mason BMCE Manchester April 2010 The Open University Maths Dept University of Oxford Dept of Education Promoting.
Improving learning in mathematics PD2: Learning from mistakes and misconceptions.
RATIONAL NUMBERS NATURAL NUMBERS {1, 2, 3,…}+ {0} = WHOLE NUMBERS {0, 1, 2, 3,…}+ NEG. WHOLE NOS. = INTEGERS {…-2, -1, 0, 1, 2,…}+ PURE FRACTIONS = RATIONAL.
THIRD AND FOURTH GRADE NUMBER AND OPERATIONS: FRACTIONS
{8, 16, 24, 32, …} are the multiples of 8 Common Multiples
What is the Sequence? There are 6 different sequential numbers
The Evolution of the Number Bond. Composing and Decomposing Numbers  “Focusing on a quantity in terms of its parts has important implications for developing.
Why Fractions? “Understanding fractions is one of the most important outcomes of mathematics education because of the pervasive use of fractions throughout.
1 Reasoning Reasonably in Mathematics John Mason Matematikbiennalen Umeä Sweden 2014 The Open University Maths Dept University of Oxford Dept of Education.
Section 2 Systems of Professional Learning Module 2 Grades K-5: Focus on Content Standards.
Multiply with decimals
Algebra Problems… Solutions Algebra Problems… Solutions © 2007 Herbert I. Gross Set 24 By Herbert I. Gross and Richard A. Medeiros next.
1 Using Mathematical Structure to Inform Pedagogy Anne Watson & John Mason NZAMT July 2015 The Open University Maths Dept University of Oxford Dept of.
Sunnyside School District
1 On the Structure of Attention & its Role in Engagement & the Assessment of Progress John Mason Oxford PGCE April 2012 The Open University Maths Dept.
Decomposing Fractions
Developing Reasoning, Explanation and Fluency St Helens Teaching Alliance Tara Loughran
Divide with decimals Standard UW.GAP.5.M.NBT.07 Add, subtract, multiply, and divide decimals to hundredths using concrete models or drawing and strategies.
Differentiated Instruction in the Primary Mathematics Classroom J. Silva.
1 While you are waiting: Warm Up 1: in a certain club there are 47 people altogether, of whom 31 are poets and 29 are painters. How many are both? Warm.
1 Reasoning Reasonably in Mathematics John Mason EARCOME 6 Phuket 2013 The Open University Maths Dept University of Oxford Dept of Education Promoting.
Sunnyside School District Math Training Module 6 Conceptual Lessons.
Comenius Math and Science Studio. Funny math How is it possible?
Unit 7 Math Review Fractions. Estimate whether the sum is closer to 0, 1/2, or 1? 2/6 + 1/6.
Diagnosing Mathematical Errors: Fractions and Decimals: Addition and Subtraction Dr. Jill Drake College of Education.
Fractions Review. Fractions A number in the form Numerator Denominator Or N D.
Teaching to the Big Ideas K - 3. Getting to 20 You are on a number line. You can jump however you want as long as you always take the same size jump.
MULTIPLYING AND DIVIDING FRACTIONS. FRACTION REVIEW.
2.1 Properties and Operations
Questioning in Mathematics Anne Watson Cayman Islands Webinar, 2013.
Big Book about Fractions By Mrs. Bader. The important thing about fractions is that they are parts of a whole. Fractions are small parts of something.
1 Geometry at Kings John Mason Dec 2015 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking.
1 Reasoning Reasonably in Mathematics John Mason Schools Network Warwick June 2012 The Open University Maths Dept University of Oxford Dept of Education.
1 Thinking Algebraically & Geometrically John Mason University of Iceland Reykjavik 2008.
1 Reasoning Masterfully Mathematically (mostly without arithmetic) John Mason Affinity Leicestershire Feb 2016 The Open University Maths Dept University.
Developing mathematical thinking in the core curriculum Anne Watson East London Maths Forum June 2008.
By: Colleen Mulcahy, Susan Hinerman, Misty Thompson, Ginger Ross, and Kelly Carney.
Woodfall Primary School Curriculum Event 2014 Changes to Mathematics.
Progression of Arrays.
1 Reasoning Reasonably in Mathematics John Mason SEMAT Køge Denmark March 2014 The Open University Maths Dept University of Oxford Dept of Education Promoting.
1 Joined Up Reflections on Drawing to a Close John Mason April 2008.
Multiply with decimals
Working Mathematically with Students Part C
PD2: Learning from mistakes and misconceptions
OR a RATIONAL NUMBER is any number that can be written as a fraction.
OR a RATIONAL NUMBER is any number that can be written as a fraction.
PD2: Learning from mistakes and misconceptions
Presentation transcript:

1 Teaching as The Directing of Attention and Learning as the Transformation of Attention 2007 John Mason Stowarzyszenia Nauczycieli Matematyki Muszyna

2 Outline: teaching using attention  Directing Attention Towards: –stressing, focusing –confronting  Directing Attention Away: –de-emphasising through exploring (integration through subordination)  Structure of Attention

3 Directing Attention Towards

4 What’s The Difference? What could be varied? –= First, add one to each First, add one to the larger and subtract one from the smaller What difference would that make to the differences? Directing Attention Towards

s are in one fifth? s are in How many... are in...? How many s are in 12 1 ? 2 ? 3 ? 4 ? blah ? oop ?... ? What are you doing? one third? 23 ? s are in Directing Attention Towards5

fraction which divides into both an integral number of times. Find a … and and another find the largest. 6 Directing Attention Towards

Find a... fraction which both divide into an integral number of times and and another find the smallest. Directing Attention Towards7

8 More doesn’t always go with More and Less doesn’t always go with Less  Draw a (non-square) rectangle –Now adjust it so that the area goes up while the perimeter  goes up  stays the same  goes down Adjust = cut &/or paste Adjust = alter rectangle Area Perimeter UpSameDown Up Same Down Dina Tirosh 2007

9 Directing Attention Away (ignoring, de-emphasising; exploring)  In how many different ways can a unit fraction be written as the difference of two unit fractions?  In how many different ways can you decompose a given rectilinear region into non-overlapping rectangles? Directing Attention Away

10 Learner Choice: Breadth & Depth Exercises 3y = 12 – 2x x = 3 2x + 3y = 12 7x – y = 19 2x + 3y = 12 21x – 3y = 57 2x + 3y = 12 23x = 69 3y = 13 – 2x x = 5 2x + 3y = 13 9x – 6y = 39 2x + 3y = 13 11x – 3y = 52 2x + 3y = 13 13x = 65 2y = x x = -2 -5x + 2y = 14 8x - y = x + 2y = 14 16x – 2y = x + 2y = 14 11x = -22 5y = x x = 2 -3x + 5y = 14 4x + y = 12 -3x + 5y = 14 20x + 5y = 60 -3x + 5y = 14 23x = 46 (Dave Herring 2007)

11 Using Attention

True or False What do you do to decide? What might 8yr olds do to decide? Marta Mollina (Granada) = = = = = 5 12 Directing Attention Towards

Raise Your Hand When You See... something which is 2/5 of something; 3/5 of something; 5/2 of something; 5/3 of something; 2/5 of 5/2 of something; 3/5 of 5/3 of something; 5/2 of 2/5 of something; 5/3 of 3/5 of something; 1 ÷ 2/5 of something; 1 ÷ 3/5 of something; 2/5 ÷ 3/5 of something? 3/4 ÷ 5/4 of something? something which is 2/5 of something; 3/5 of something; 5/2 of something; 5/3 of something; 2/5 of 5/2 of something; 3/5 of 5/3 of something; 5/2 of 2/5 of something; 5/3 of 3/5 of something; 1 ÷ 2/5 of something; 1 ÷ 3/5 of something; 2/5 ÷ 3/5 of something? 3/4 ÷ 5/4 of something? 13 Pat Thompson (2004) Directing Attention Towards

Ride & Tie Which graph makes sense, and why? What is the same, and what different about Directing Attention Towards

Structures of Attention Macro Structures ★ locus, focus, multiplicity Micro Structure ★ Holding Wholes (gazing) ★ Discerning Details ★ Recognising Relationships (situated; particular) ★ Perceiving Properties (general) ★ Reasoning on the basis of stated properties Macro Structures ★ locus, focus, multiplicity Micro Structure ★ Holding Wholes (gazing) ★ Discerning Details ★ Recognising Relationships (situated; particular) ★ Perceiving Properties (general) ★ Reasoning on the basis of stated properties 15

Structures of Attention Macro Structures ★ locus, focus, multiplicity Micro Structure ★ Holding Wholes (gazing) ★ Discerning Details ★ Recognising Relationships (situated; particular) ★ Perceiving Properties (general) ★ Reasoning on the basis of stated properties Macro Structures ★ locus, focus, multiplicity Micro Structure ★ Holding Wholes (gazing) ★ Discerning Details ★ Recognising Relationships (situated; particular) ★ Perceiving Properties (general) ★ Reasoning on the basis of stated properties 16

17 Magic Square Reasoning –= 0Sum( )Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2 Directing Attention Towards

18 More Magic Square Reasoning –= 0Sum( )Sum( ) Directing Attention Towards

Questions ★ Why is directing attention important? Need for confluence between learner and teacher attention Being aware of the structure and focus of your own attention is necessary if you are to direct the attention of learners appropriately Being aware of the importance of attention can alert you to reasons for what learners say and do ★ Why is directing attention important? Need for confluence between learner and teacher attention Being aware of the structure and focus of your own attention is necessary if you are to direct the attention of learners appropriately Being aware of the importance of attention can alert you to reasons for what learners say and do 19

Summary ★ ★ How is attention directed? Towards (stressing, focusing) Towards (stressing, focusing) Away (ignoring, de-emphasising) Away (ignoring, de-emphasising) Strategies for Directing Attention Say What You See Watch What You Do In How Many Different Ways Can You...? Show me that you see... What is the same, and what is different about … ? Making Use of the Structure of Attention ★ ★ How is attention directed? Towards (stressing, focusing) Towards (stressing, focusing) Away (ignoring, de-emphasising) Away (ignoring, de-emphasising) Strategies for Directing Attention Say What You See Watch What You Do In How Many Different Ways Can You...? Show me that you see... What is the same, and what is different about … ? Making Use of the Structure of Attention