Insights into mathematics teaching methods based on variation Anne Watson amp J ohn Mason MAST cohort 8 Edge Hill Ormskirk June 2017 Promoting Mathematical Thinking University of Oxford ID: 628298
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What can vary and what stays the same? Insights into mathematics teaching methods based on variation
Anne Watson & John MasonMAST cohort 8Edge Hill Ormskirk June 2017
Promoting Mathematical Thinking
University of Oxford
Dept of Education
The Open University
Maths DeptSlide2
Ways of WorkingWe work through lived experience: ours and yoursWe offer tasks for you, to work on with colleagues2Slide3
Plan for this workshopDirect experience of the use of variation in thinking about measureVariation principles and more examplesPlanning variation and invarianceFrom pattern spotting to recognising relationships3Slide4
Measuring LengthYou have been given a paper stripPlease measure the length of the strip using various rods and record the measurement in the table provided4Slide5
Colour of rod
Number of rodsL = length of (colour) rod x (number)
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Colour of rod
Number of rodsL = length of (colour) rod x (number)
pink
9
L = length of pink x 9
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Colour of rod
Number of rodsL = length of (colour) rod x (number)
white
red
light green
pink
yellow
dark green
black
brown
blue
orange
What relationships can be detected?
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Further VariationWhat if the line was 72 units? the line was 18 units? the line was 144 units? the line was 12 units?What if the white rod represents 1m? the yellow rod represents 1m?…
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Variation PrinciplesSwedish:We notice a feature that varies against a background of invarianceWhat is available to be learned (about)is what has been variedOur variation:We may notice what is constant within a varying context
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Find a number half way between: 28 and 34 2.8 and 3.4 38 and 44 -34 and -28 9028 and 9034 .0058 and .0064
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Variation theoriesChinese:One problem multiple methods of solution (OPMS)One problem multiple changes (OPMC)Multiple problems one solution method (MPOS) (Procedural and conceptual) (Static and dynamic)
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Analysing variation and invariance in the taskmeasuring unitexact or not exact length of line given in units < cms.- length of line given in units > cms.
actual length of linemeasuring method, repeated unitsformat for resultsformat for number sentenceVariationInvarianceSlide14
ConsiderationsIntended / enacted / lived object of learningAuthor intentionsTeacher intentionsLearner experience
Task
Author intentions
Teacher intentions
As presentedAs interpreted by learnersWhat learners actually attemptWhat learners actually doWhat learners experience and internaliseDidactic Transposition
Expert awareness
is transformed into
instruction in behaviour14Slide15
The Drakensberg Grid15Slide16
Selected Columns16Slide17
Selected Row17Slide18
Single row expanded18Slide19
Giant19Slide20
Role of formatting to draw attention to variation and invariance
objectg
÷
h
=r
g
÷
r
=
h
h
x
r
=
g
shoelace
bus pass width
footprint
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ReflectionIntended enacted lived object of learningUse of variation to bring lived object of learning and intended object of learning together
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Reflection on the effects of variation on youWhat struck you during this session?What for you were the main points to think about (cognition)?What upset you or got you going (affect)?What actions might you want to use in your teaching and talk about with others (awareness) ?
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Availability of variation/invariant relationVisual, available without teacher directionVisual, available with teacher directionVisual or non-visual and independent of prior knowledgeVisual or non-visual but dependent on prior knowledgeDependent on prior knowledge and teacher direction and choice
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Does Variation Principle (VP) bring something to maths that cannot be seen already?Maths is about variation/invarianceVP gives focus, language, structureVP commits you to analysing and constructing variation as a professional toolExperiential, no need for ‘black box’ e.g neuroscience; laboratory studies
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Role in English policy and practiceShanghaiNCETM training and videosTextbook designProfessional training and developmentThe next big thing…......? 25Slide26
Variation used in teachingBe clear about the intended concept to be learned, and work out how it can be experienced through varied examplesMatching up varied representations of the same example helps learningThe intended object of learning is often an abstract relationship that can only be experienced through examples
When a change in one variable causes a change in another, learners need several well-organised examples and reflection to ‘see’ relation and structureVariation of appropriate dimensions can sometimes be directly visible, such as through geometry or through page layoutDraw attention to connections, similarities and differencesUse deep understanding of the underlying mathematical principles26Slide27
Follow-UpPMTheta.comThinkers (ATM)Questions & Prompts for Mathematical Thinking (Primary) (ATM)27