Presentation on theme: "Reasoning Mathematically"— Presentation transcript:
1 Reasoning Mathematically Promoting Mathematical ThinkingReasoning MathematicallyJohn MasonMathsfest CorkOct 2012The Open UniversityMaths DeptUniversity of OxfordDept of Education
2 Specific Aims for Ordinary Level an understanding of mathematical concepts and of their relationshipsconfidence and competence in basic skillsthe ability to solve problemsan introduction to the idea of logical argumentappreciation both of the intrinsic interest of mathematics and of its usefulness and efficiency for formulating and solving problems
3 ConjecturesEverything said here today is a conjecture … to be tested in your experienceThe best way to sensitise yourself to learnersis to experience parallel phenomena yourselfSo, what you get from this session is what you notice happening inside you!
4 TasksTasks 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 aloneSomething more is required
5 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 wayCold means that it is at least two places awayWhat is your best strategy to locate the secret place?
6 Counting OutIn 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?ABCDE12345How do you know?Generalise!987610…If that object is eliminated, you start again from the ‘next’. Which object is the last one left?
7 Alternating Square Sums Imagine a triangleImagine a point inside the triangleDrop perpendiculars to the three sides of the triangleEach side of the triangle comprises two segmentsOn each segment of each edge, construct a squareAlternately colour the squares yellow and cyan around the triangleConjecture: 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?
8 Selective SumsAdd up any 4 entries, one taken from each row and each column.The answer is (always) 6Why?-22-4648315-1-3Example of (use of) permutationsUse fractions to urge practice with fractionsGet students to make up their own with their age or some other number as the answerExample of seeking invariant relationshipsExample of focusing on actions preserving an invarianceOpportunity to generalise
9 Selective SumsAdd up any 4 entries, one taken from each row and each column.Is the answer always the same?Why?
10 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?2-5-3-64-193-25Why 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?
11 Circles in Circles How are the red and yellow areas related? red orangeyellow
12 Carpet TheoremsIn a room there are two carpets whose combined area is the area of the room.The area of overlap is the area of floor uncoveredIn 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
13 Rectangular Room with 2 Carpets How are the red and blue areas related?
14 Perimeter Projections The red point traverses the quadrilateralThe 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?
15 Could these all be squares? Square DeductionCould these all be squares?
16 Square Deduction: tracking arithmetic (3x3+4)/33x4-3x33+3x43x3+4343+2x42x3+43+4Track the 3 and the 4:Replace the 3 by a and the 4 by b
17 Square Deduction: acknowledging ignorance (3a+b)/33a+b = 3(3b-3a)12a = 8bSo 3a = 2b3b-3aa+3b3a+babFor an overall square4a + 4b = 2a + 5bSo 2a = ba+2b2a+ba+b23587911For n squares upper leftn(3b - 3a) = 3a + bSo 3a(n + 1) = b(3n - 1)But not also 2a = b
18 Reflection What aspects of reasoning… Stood out for you? Involved some struggleWhat actions …Did you undertake?Were ineffective (why?)Were effective (why?)
19 TasksTasks 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 aloneIt is not the task that is rich …but whether it is used richlyWhat matters more than the particular answer is …how do you know?what can you vary and still the same approach works?
20 Reminder Do–Talk–Record Generating need to communicate Provoking EngagementSurpriseChallenge (trust)Promoting Mathematical thinkingHow do you know? …Why must …?Active StudentsConstructingExtending
21 Institute of Mathematical Pedagogy Follow Upmcs.open.ac.uk/jhm3open.ac.ukThinking Mathematically (new edition)Designing and Using Mathematical TasksQuestions and Prompts … (Primary version from ATM)ThinkersInstitute of Mathematical PedagogyAugustmcs.open.ac.uk/jhm3