Key questions

• ? Relationship between goal-directed activities and exploratory activities
• ? Integration of individual artefacts
• ? Relationship between concrete artefacts and abstract concepts of mathematics

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? Relationship between goal-directed activities and exploratory activities

What does exploratory mean?

what does SBR mean by learning more linear algebra every time he teaches CS131?
cf. "knowing each other better" isn't knowing your full name age, address, height etc
making connections with other elements of experience

to what extent does becoming very proficient at multiplying matrices involve learning?
cf. the role of technical exercises in learning to play a musical instrument

does the specification of a recipe necessarily admit the possibility of "standing and staring"
"what is this life if, full of care, we have no time to stand and stare"
busy/preoccupied/absorbed not as in doing something specific, but as in savouring the present

'neighbouring in the space of sense' - cf. unlikely facts about a person "my father / my son had / has red hair'

CS in fact favours specification of recipes that are based on oblivious agent action (hence efficient)

exploratory activity here means anything but being 'oblivious' to context
rather - acutely sensitive to context - open to new meanings - actively seeking new connections
provoking questions - negotiating common understandings

cf. inquiry-based learning, regulating confusion ...

contrast 'is this the right way?' with "is this an enriching place?"

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? Integration of individual artefacts

Perspective that seems to get endorsement from SBR's Lego analogy / Joe's optimism re OO software development
- but if we can signpost and spell out then to some degree are limited
- of course, 'limited' doesn't mean necessarily no longer affording interpretation (cf. definitive notations in tkeden)
- excluding some opportunistic re-interpretation cf. way in which text highlights in "SBR's Lego analogy / Joe's"

Integration in mind in EM is conjunction of experiences - commonplace of experience
cf. attention to the whole group vs "individual attention"

cf. conceptual integrity in sw (Fred Brookes)

EFL - integration of perspectives as a feature of convergence to the objective world

creating a common stage for interaction - but not constraining the individual interactions to the commonplace
- affording the idiosyncratic

e.g. let's conjoin JUGS with Joe's construal

distinction between 're-writing' and assimilating through introducing dependencies
the merits of both disjunction and conjunction cf. Dave West's "maximising cohesion whilst minimising coupling"
need ways in which to isolate the individual agent perspectives whilst at the same time integrating them

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? Relationship between concrete artefacts and abstract concepts of mathematics

cf. Euclidean lines and concrete imperfect lines on paper
numbers we can experience vs numbers we can calculate with / estimate / compute only with ingenuity

abstract concepts can lie behind many concrete artefacts - "from pure to applied mathematics" (cf. LMS/IMA)
but (more pertinent to EM) concrete artefacts inform abstract concepts (cf. James - all 'knowing' as rooted in conjunction)

making rich connections between abstract concepts and concrete experiences is significant learning
'concrete' here need not mean 'applicable', as in relating to some 'real-world' context
many things are concrete in my immediate experience and yours that do not belong to our objective world
- issue is one of reliability, repeatability, consistency

cf. James classification of experiences into subjective and objective not an intrinsic quality of the experience
- consider what is culturally acceptable

limits on what is accessible by way of concrete experience ...

technology: resource (blackboard, whiteboard, computer, wiimote),
notation (JS-EDEN, tkeden, html, SVG)

personal skill: ability to realise/register conjunctions relates to previous experience / practice

cultural: common language e.g. English, common conventions (underlining vectors, using boldface, using upper-lower case)
how you layout matrices and vectors (cf. Einstein summation notation)

... where to draw the line between the accidental and the essential?

An important issue is to what extent this agenda is motivated. Why consider this for learning?

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Appendix

Joe extension

pt is [lambda, mu];
transpt is [lambda * PcoefA + mu * PcoefB, lambda * QcoefA + mu * QcoefB];
Opt is Line(540,240,lambda, mu, "yellow");
Otranspt is Line(40,240,transpt[1], transpt[2], "grey");

picture is [MagQtext, MagPtext, PdotQText, AcuteAngPQText, AnglePOQText, AnglePQ, buttonA, buttonB, baseAHead, baseBHead, buttonO, plusSymbQ, eqSymbQ, plusSymbP, eqSymbP, QcoefADiv, QcoefBDiv, PcoefBDiv, PcoefADiv, vectorBrack6, vectorBrack5, vecX5, vecX6, vecY5, vecY6, baseText3, baseText4, vectorBrack2, vecX1, vecX2, vecY1, vecY2, ColumnText1, ColumnText2, buttonReset, buttonSnapQO, buttonSnapPO, buttonSnapPQ, buttonSnapQP, vector_p_text, vector_q_text, vectorActualResult, vectorActualText, vectorResultText, vectorResultAline, vectorResultBline, vectorResultHead, pointQ, vectorQ, vectorQtext, alert, vecX4, vecY4, vectorBrack4, vectorBrack3, resultText, vecY3, vecX3, baseText2, vectorBrack1, baseText, baseAvector, baseAtext, baseBvector, baseBtext, origin, originText, pointP, vectorP, vectorPtext, buttonP, buttonQ, Opt, Otranspt];

Making an arrow (of size proportional to the length of the vector)

a = 300;
b = 300;
l is Line(a,b,x,y,"green");
x = 600; y = 400;
mu = 0.9;
ptarr is [a + 0.9 * (x-a), b + 0.9 * (y-b)];
## orthl is Line(ptarr[1], ptarr[2], ptarr[1] + (b-y), ptarr[2] + (x-a), "red");
## picture is [l, orthl];
larr is Line( ptarr[1] + (b-y) * (1-mu), ptarr[2] + (x-a) * (1-mu), x, y, "green");
rarr is Line( ptarr[1] + (b-y) * (mu-1), ptarr[2] + (x-a) * (mu-1), x, y, "green");
picture is [l, larr, rarr];

mvX is Slider("mvX", 100, 500, 5, 300, "horizontal", 300, 10);
mvY is Slider("mvY", 100, 500, 5, 300, "vertical", 10, 300);
picture is [l, larr, rarr, mvX, mvY];

x is mvX_value; y is mvY_value;

Exercise: make size of arrow invariant, make a solid arrow head. Possibility of making a JavaScript arrow "object"?

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