Presentation on theme: "Promoting Mathematical Thinking"— Presentation transcript:
1 Promoting Mathematical Thinking Task Construction: lessons learned from 25 years of distance support for teachersJohn MasonNottinghamFeb 2012The Open UniversityMaths DeptUniversity of OxfordDept of Education
2 Outline Some Tasks OU Frameworks MGA, DTR, Stuck, EIS, … APC or ORA: Own experience, Reflection on parallels, Apply to classroomSystematics FrameworksWhat makes a task ‘rich’?
3 Number Line Translations… Imagine a number line with the integers marked on itImagine a copy of the number line sitting on top of itTranslate the copy line to the right by 3I am thinking of a number … tell me how to work out where it ends upWhere does 7 end up?Where does –2 end up?12345678-1-2-3-4-5-6-7Chosen this because I have always enjoyed it, it is richly mathematical, and because of recent discussions with an OU studentParticular to GeneralDenote translation to the right by a, by TaWhat is Ta followed by Tb?What about Tb followed by Ta?
4 Number Line Scaling…Imagine a number line with the integers marked on itImagine a copy of the number line sitting on top of it12345678-1-2-3-4-5-6-7Denote scaling from 0 by a factor of s by SsI am thinking of a number … tell me how to work out where it ends upGeneral to particularWhat is Sa followed by Sb?Denote scaling from p by a factor of s by Sp,sWhat is Sp,s in terms of T and Ss?
5 Number Line Scaling…Imagine a number line with the integers marked on itImagine a copy of the number line sitting on top of itScale the number line by a factor of 3(keeping 0 fixed)12345678-1-2-3-4-5-6-7I am thinking of a number … tell me how to work out where it ends upWhere does 2 end up?Where does –3 end up?Particular to generalDenote scaling from 0 by a factor of s by SsWhat is Sa followed by Sb?Denote scaling from p by a factor of s by Sp,sWhat is Sp,s in terms of T and Ss?
6 Number Line Rotations… Imagine a number line with the integers marked on itImagine a copy of the number line sitting on top of itRotate the copy through 180° about the point 312345678-1-2-3-4-5-6-7I am thinking of a number … tell me how to work out where it ends upWhere does 7 end up?Where does -2 end up?Denote rotating about the original point p by RpWhat is Rp followed by Rq?Rotate twice about 0 …… to see why R–1R–1 = T0 = S1 and so (-1) x (-1) = 1
7 Diamond Multiplication Chosen this because it always provokes immediate ‘work’ by audiences; illustrates beautifully the movement of attention
8 Differing Sums of Products Write down four numbers in a 2 by 2 grid4537Add together the products along the rows= 43Add together the products down the columns= 4143 – 41 = 2Calculate the differenceTracking ArithmeticThat is the ‘doing’ What is an undoing?What other grids will give the answer 2?Choose positive numbers so that the difference is 7
9 Differing Sums & Products 4537Tracking Arithmetic4x7 + 5x34x5 + 7x34x(7–5) + (5–7)x3= 4x(7–5) – (7–5)x3= (4-3) x (7–5)So in how many essentially different ways can 2 be the difference?What about 7?Revealing Structure by attending to relationships not calculationsNumerals as placeholdersSo in how many essentially different ways can n be the difference?
10 Patterns with 2Embedded Practice(Gattegno & Hewitt)
12 Put your hand up when you can see … Something that is 3/5 of something elseSomething that is 2/5 of something elseSomething that is 2/3 of something elseSomething that is 5/3 of something else…Something that is 1/4 – 1/5 of something else
13 Remainders What is the remainder on dividing 5 by 3? What question am I going to ask next?What is the remainder on dividing 5 by -3?What is the remainder on dividing -5 by -3?
14 Task Purposes To introduce or extend contact with concepts To highlight awareness of human powers used mathematicallyTo focus attention on mathematical themesTo sharpen awareness ofstudy strategiesproblem solving strategies (heuristics)learning how to learn mathematicsevaluating own progressexam techniquePurpose for studentsPotential Utility(Ainley & Pratt)
15 Learning from TasksTasks –> Activity –> Actions –> ExperienceBut one thing we don’t seem to learn from experience …is that we don’t often learn from experience alone!–> withdraw from action and reflect upon itWhat was striking about the activity?What was effective and what ineffective?What like to have come-to-mind in the future?Personal propensities & dispositions?Habitual behaviour and desired behaviour?Fresh or freshened awarenesses & realisations?
16 Task Design Pre-paration Pre-flection Post-paration Post-flection Content(Mathematics)ReflectionInteractions(as transformative actions)TasksResourcesActivityWhen does learning take place?In sleep!!!
17 SlogansA lesson without opportunity for learners … to generalise mathematically … is not a mathematics lesson! to make and modify conjectures; to construct a narrative about what they have been doing; to use and develop their own powers; to encounter pervasive mathematical themes is not an effective mathematics lesson Trying to do for learners only what they cannot yet do for themselves
18 Modes of interaction Expounding Explaining Exploring Examining ExercisingExpressing
21 Potential What builds on it (where it is going) Most it could be Math’l & Ped’c essenceRole of reflection in achieving affordancesLeast it can beWhat it builds on(previous experiences)Affordances– Constraints–Requirements(Gibson)Directed–Prompted–SpontaneousScaffolding & Fading (Brown et al)ZPD (Vygotsky)
22 Thinking Mathematically CMEDo-Talk-Record (See–Say–Record)See-Experience-MasterManipulating–Getting-a-sense-of–ArtculatingEnactive–Iconic–SymbolicDirected–Prompted–SpontaneousStuck!: Use of Mathematical PowersMathematical Themes (and heuristics)Inner & Outer Tasks
24 Example From NNP project, pattern sequences to be counted Stuck with providing first, second, third and only later recognising the dependency createdUnlocking potentialUniversality of the Frame Theorem (Gaussian Curvature and Betti Numbers)Counting squares, counting sticks, …Counting weights
25 Example: Extending Mathematical Sequences Mathematically “What is the next term …?” only makes sense when ...Mathmematical guarantee of uniquenessGeometrical or other construction sourceSome other constraint
26 Painted Wheel (Tom O’Brien) …Someone has made a simple pattern of coloured squares, and then repeated it at least once moreState in words what you think the original pattern wasPredict the colour of the 100th square and the position of the 100th white square…Theorem: a sequence is uniquely specified if you know that the repeating pattern has appeared at least twice.Make up your own:a really simple onea really hard oneProvide two or more sequences in parallel
27 Gnomon Border How many tiles are needed to surround the 137th gnomon? The fifth is shown hereIn how many different ways can you count them?What shapes will have the same Border Numbers?
28 Extending Mathemtical Sequences Stress in Thinking Mathematically and later on ‘specifying the growth mechanism before trying to count things’Uniquely Extendable Sequences TheoremInstance of general topological theorem (Betti numbers)Attempts in two Dimensions!
29 How many holes for a sheet of r rows and c columns PerforationsIf someone claimed there were 228 perforations in a sheet, how could you check?How many holes for a sheet of r rows and c columnsof stamps?
31 Attention Teahing and Learning is fundamentally about attention: What is available or likely to come–to–mind when neededWhat is available to be learned?variationUse of powersUse of themesUse of resources (physical, mental, virtual)Structure of attentionHolding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties
32 Follow-Up j.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3 Designing & Using Mathematical Tasks (Tarquin/QED)Thinking Mathematically (Pearson)Developing Thinking in Algebra, Geometry, Statistics (Sage)Fundamental Constructs in Mathematics Education (RoutledgeFalmer)Mathematics Teaching Practice: a guide for university and college lecturers (Horwood Publishing)Mathematics as a Constructive Activity (Erlbaum)Questions & Prompts for Mathematical Thinking (ATM)Thinkers (ATM)Learning & Doing Mathematics (Tarquin)open.ac.ukmcs.open.ac.uk/jhm3