Reasoning Mathematically Promoting Mathematical Thinking Reasoning Mathematically John Mason Mathsfest Cork Oct 2012 The Open University Maths Dept University of Oxford Dept of Education

Specific Aims for Ordinary Level an understanding of mathematical concepts and of their relationships confidence and competence in basic skills the ability to solve problems an introduction to the idea of logical argument appreciation both of the intrinsic interest of mathematics and of its usefulness and efficiency for formulating and solving problems

Conjectures Everything said here today is a conjecture … to be tested in your experience The best way to sensitise yourself to learners is to experience parallel phenomena yourself So, what you get from this session is what you notice happening inside you!

Tasks Tasks promote activity; activity involves actions; actions generate experience; but one thing we don’t learn from experience is that we don’t often learn from experience alone Something more is required

Secret Places What is your best strategy to locate the secret place? Homage to Tom O’Brien (1938 – 2010) One of the places around the table is a secret place. If you click near a place, the colour will tell you whether you are hot or cold: Hot means that the secret place is within one place either way Cold means that it is at least two places away What is your best strategy to locate the secret place?

Counting Out In a selection ‘game’ you start at the left and count forwards and backwards until you get to a specified number (say 37 or 177). Which object will you end on? A B C D E 1 2 3 4 5 How do you know? Generalise! 9 8 7 6 10 … If that object is eliminated, you start again from the ‘next’. Which object is the last one left?

Alternating Square Sums Imagine a triangle Imagine a point inside the triangle Drop perpendiculars to the three sides of the triangle Each side of the triangle comprises two segments On each segment of each edge, construct a square Alternately colour the squares yellow and cyan around the triangle Conjecture: the sum of the areas of the yellow squares is the sum of the areas of the cyan squares. For what hexagons is this the case?

Selective Sums Add up any 4 entries, one taken from each row and each column. The answer is (always) 6 Why?   -2 2 -4 6 4 8 3 1 5 -1 -3 Example of (use of) permutations Use fractions to urge practice with fractions Get students to make up their own with their age or some other number as the answer Example of seeking invariant relationships Example of focusing on actions preserving an invariance Opportunity to generalise

Selective Sums   Add up any 4 entries, one taken from each row and each column. Is the answer always the same? Why?

Chequered Selective Sums Choose one cell in each row and column. Add the entries in the dark shaded cells and subtract the entries in the light shaded cells. What properties makes the answer invariant? What property is sufficient to make the answer invariant?   2 -5 -3 -6 4 -1 9 3 -2 5 Why is it suffient that each 2 by 2 subsquare sums to zero for the property to hold? How does this grid relate to a Covered up Sum grid?

Circles in Circles How are the red and yellow areas related? red orange yellow

Carpet Theorems In a room there are two carpets whose combined area is the area of the room. The area of overlap is the area of floor uncovered In a room there are two carpets. They are moved so as to change the amount of overlap. The change in the area of overlap is the change in area of uncovered floor

Rectangular Room with 2 Carpets How are the red and blue areas related?

Perimeter Projections The red point traverses the quadrilateral The vertical movement of the red point is tracked. What shape is the graph? Given a graphical track of the vertical movement and the horizontal movement, What is the shape of the polygon?

Could these all be squares? Square Deduction Could these all be squares?

Square Deduction: tracking arithmetic (3x3+4)/3 3x4-3x3 3+3x4 3x3+4 3 4 3+2x4 2x3+4 3+4 Track the 3 and the 4: Replace the 3 by a and the 4 by b

Square Deduction: acknowledging ignorance (3a+b)/3 3a+b = 3(3b-3a) 12a = 8b So 3a = 2b 3b-3a a+3b 3a+b a b For an overall square 4a + 4b = 2a + 5b So 2a = b a+2b 2a+b a+b 2 3 5 8 7 9 11 For n squares upper left n(3b - 3a) = 3a + b So 3a(n + 1) = b(3n - 1) But not also 2a = b

Reflection What aspects of reasoning… Stood out for you? Involved some struggle What actions … Did you undertake? Were ineffective (why?) Were effective (why?)

Tasks Tasks promote activity; activity involves actions; actions generate experience; but one thing we don’t learn from experience is that we don’t often learn from experience alone It is not the task that is rich … but whether it is used richly What matters more than the particular answer is … how do you know? what can you vary and still the same approach works?

Reminder Do–Talk–Record Generating need to communicate Provoking Engagement Surprise Challenge (trust) Promoting Mathematical thinking How do you know? … Why must …? Active Students Constructing Extending

Institute of Mathematical Pedagogy Follow Up mcs.open.ac.uk/jhm3 j.h.mason @ open.ac.uk Thinking Mathematically (new edition) Designing and Using Mathematical Tasks Questions and Prompts … (Primary version from ATM) Thinkers Institute of Mathematical Pedagogy August 6-9 2013 mcs.open.ac.uk/jhm3