1 / 21

Using Theoretical Constructs to Inform Teaching

Using Theoretical Constructs to Inform Teaching. John Mason IMEC9 Sept 2007. Outline. Teaching Mathematics Tasks, activities, experience, reflection Teaching People To Teach Mathematics Consistency Awareness of the role of Tasks, activities, experience, reflection. My Methods.

chadwick-mills
Download Presentation

Using Theoretical Constructs to Inform Teaching

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Using Theoretical Constructs to Inform Teaching John Mason IMEC9 Sept 2007

  2. Outline • Teaching Mathematics • Tasks, activities, experience, reflection • Teaching People To Teach Mathematics • Consistency • Awareness of the role of • Tasks, activities, experience, reflection

  3. My Methods The canal may not itself drink, but it performs the function of conveying water to the thirsty • Experiential What you get from this session iswhat you noticehappens inside you you, and how you relate that to your own situation • Reflection • Linking to theories • Preparing to notice more carefully in future • Brief-but-vivid accounts

  4. The square of the larger added to the smaller? • The square of the smaller added to the larger? Don’t calculate!!! Conjecture! Only then Check! One Sum • I have two numbers which sum to 1 • Which will be larger:

  5. a a One Sum Diagrams 1 (1-a)2 1 1-a a2 Anticipating,not waiting

  6. 2.499…97 2.497 2.479…9 2.479 Decimal • Write down a decimal number between 2 and 3 • and which does NOT use the digit 5 • which DOES use the digit 7 • and which is as close to 5/2 as possible 2.47 2.49…979…

  7. Difference of Two • Write down two numbers which differ by 2 • And another pair • And another pair • And another pair which obscure the fact that the difference is 2 Fractions? Decimals? 9999 & 10001 Negatives? … ?

  8. Characterising • What numbers can be two more than the sum of four consecutive whole numbers? What numbers can be one more than the product of four consecutive numbers? What did you do first? Do you encourage your learners to do this? How often do you set tasks for themwhere they need to do this?

  9. Sketchy Graphs Sketch the graphs of a pair of straight lines whose y-intercepts differ by 2 Sketch the graphs of a pair of straight lineswhose x-intercepts differ by 2 Sketch the graphs of a pair of straight lineswhose slopes differ by 2 Sketch the graphs of a pair of straight linesmeeting all three conditions

  10. area more same less altitude more more altless area more altmore area more altsame area Same altmore area same altless area same less less altmore area less altsame area less altless area More Or Less Altitude & Area Draw a scalene triangle

  11. area more same less perimeter more more perimless area more perimmore area more perimsame area Same perimmore area same perimless area same less less perimmore area less perimsame area less perimless area More Or Less Area & Perimeter Draw a rectangle When can it be done? When can it not be done?

  12. Omar Khayam In childhood we strove to go to school,
Our turn to teach, joyous as a rule
The end of the story is sad and cruel
From dust we came, and gone with winds cool. Pursuing knowledge in childhood we rise
Until we become masterful and wise
But if we look through the disguise
We see the ties of worldly lies Myself when young did eagerly frequent
Doctor and Saint, and heard great ArgumentAbout it and about: but evermore
Came out by the same Door as in I went

  13. MGA & DTR Doing Talking Recording

  14. Powers • Specialising & Generalising • Conjecturing & Convincing • Imagining & Expressing • Ordering & Classifying • Distinguishing & Connecting • Assenting & Asserting

  15. Themes • Doing & Undoing • Invariance Amidst Change • Freedom & Constraint • Extending & Restricting Meaning

  16. Habit forming can be habit forming One thing we do not often learn from experience, is that we do not often learn from experience alone Absence of evidenceis NOTevidence of absence A sequence of experiences does not add up to an experience of that sequence Protases

  17. Implicit Contract • If learners ‘do’ the tasks they are set, then they will ‘learn’ what is required • Contrat didactique • The more clearly and specifically the teacher specifies the behaviour sought, the easier it is for learners to display that behaviour without encountering mathematics, without thinking mathematically • Didactic tension

  18. Task & Activity • A task is what an author publishes, what a teacher intends, what learners undertake to attempt. • These are often very different • What happens is activity • Teaching happens in the interaction made possible by activity: performing familiar actions in new ways to make new actions • Learning happens through reflection and integrating new actions into functioning Teaching takes place in timeLearning takes place over time

  19. Inner World of imagery Worldof Symbols Material World Worlds of Experience enactive iconic symbolic

  20. Worlds, MGA, DTR • Enactive-Iconic-Symbolic • Three modes; three worlds • Manipulating–Getting-a-sense-of–Articulating • Doing–Talking–Recording

  21. Further Reference • Mathempedia (http://www.ncetm.org.uk) • Fundamental Constructs in Mathematics Education, RoutledgeFalmer, London (2004). • Designing and Using Mathematical Tasks. St. Albans: Tarquin. J.H.Mason@open.ac.uk http://mcs.open.ac.uk/jhm3

More Related