Friday, October 19, 2012

Capacity focus, 68c: Developing capacity to think abstractly (i.e. Piaget's Formal Operations stage) by using the CPA/CRA -- Concrete, Pictorial/ Representational, Abstract approach (and with GeoGebra as a free tool . . . )

(Education transformation series to date: CF55, a, b, c, d, e, f, CF 66 , CF 68, 68b)

It is said that Piaget would use the physical principle of moments as a test of ability to think abstractly, as to understand it one has to analyse how both the size of the force and the distance from the pivot interact jointly to yield the turning effect:

The principle of moments in action, through leverage: a smaller force, further from the pivoting point can balance or overwhelm a larger force closer to it. (BTW, that is why Archimedes said, give me a place to stand and I will move the whole world.) Where, numerically the moment, M, equals the magnitude F, multiplied by the perpendicular distance, d, of its line of action from the pivot: M = F*d. (In equilibrium, clockwise and anticlockwise moments will be equal about any point, which can be used to deduce forces etc. Don't forget the force exerted by the weight of the "beam" and the natural pivot about the centre of gravity.) [Source:, UK]
In short, the joint, direct variation of turning effect with force (F) and distance from  pivot (d) shows that there is more to using a crowbar or a hammer than may be obvious.

Which is just the point. 

If we are to understand a great many things, many of which are quite significant, we need to understand abstraction and be able to juggle multiple effects or facets in our minds. But only perhaps about up to 1/3 of adults in advanced countries can be counted on to do so on an arbitrary topic. And where children from about 11 - 14 on will increasingly need to be able to think and manipulate objects, ideas and symbols abstractly if they are to succeed at learning tasks. With of course Mathematics, Physics and Chemistry in the forefront.

Sun-and-Planets atomic model, useful for basic
Chemistry (Source: Universe Today. For more
details and more advanced points, cf. here.)
Speaking of which, I am noticing that students nowadays often do not have a reasonably clear initial picture of the atom or of molecules, such as the sun-and-planets model. And this, in a day where right from opening phases of biology, the molecular view of cell-level life processes is now going to be taught.

(This is a case where I think it useful to give an initial rough approximation, and then we can refine it through the results of quantum mechanics later on. Certainly, we can justify this through the idea that after Rutherford's alpha scattering exercise where he saw the equivalent of a Battleship's 1/2-ton+ shells bouncing back from a piece of tissue paper, the atom definitively has a hard, narrow, heavy core, of perhaps 10^-15 or so m across, and on other experiments we know atoms are about 10^-10 or so m across. That puts the electrons in orbits, and since such motion is accelerated AND the atom is stable, we see the basis for Bohr's standing waves orbits quantum model, which refines the circles by one step. That is, if the atom were a classical system the orbiting charges would spiral into the nucleus and give off electromagnetic [E-M] waves (e.g. radio, light), leading to an unstable atom. But the atom is stable and spectroscopy shows that E-M energy moves into or out of atoms in lumps, so the atom has stationary states with energy levels, which -- at first approximation -- can be viewed as orbits that form standing waves similar to how a given taut string forms standing waves of fixed frequency with nodes and antinodes. A few comments on pairing electrons in bonds and on filling up or going back down to a complete shell will help students understand a lot of chemistry and electrical behaviour. Then, when students need to understand pi bonds in organic chemistry or the like, or meet more detailed spectroscopy, the different orbitals and shapes, for s, p, d, f etc can be brought out. Of course, we should let children know there is more than this basic model or map, but it is a very useful picture. [Why not, use a map as a concrete expression of how a useful and good enough model is not going to be true in all respects . . . ?])

Ability to handle abstractions is clearly vital, especially in Math and Science.

The CPA/ CRA approach -- Concrete, Pictorial/ Representational, Abstract -- is a way for us to tackle the challenge, with some hope of at least moderate success.

We have already been seeing it in action above, twice. At least, in part. 

You will see that I did not just talk about levers or atoms, I used iconic pictures that can be riveted in our minds and memories. And in the case of crowbars and hammers, most of us will have had some concrete experience that the images will call up, down to how it feels when we pull a nail for instance.

Where I would go from there is a demonstration with a door and two students, as a class experiment: A big student pushing hard right next to the hinge -- use a solid, well-mounted door, please! -- can be countered by a slight one pushing back at the far edge, next to the knob. Then, we can reduce our findings to a mathematical model of balancing clockwise and anticlockwise moments. This can be explored experimentally with a metre stick mounted on a pivot with suspended weights. (And we may even use this to determine the upthrust on a weight suspended in water.)

Problem exercises can then be based on this. (Indeed, a great many Physics problems are versions of experimental exercises.)

The CRA/CPA approach builds on this sort of technique. 

For instance, with primary level math in view:
  • Concrete. In the concrete stage, the teacher begins instruction by modeling each mathematical concept with concrete materials (e.g., red and yellow chips, cubes, base-ten blocks, pattern blocks, fraction bars, and geometric figures).
  • Representational. In this stage, the teacher transforms the concrete model into a representational (semiconcrete) level, which may involve drawing pictures; using circles, dots, and tallies; or using stamps to imprint pictures for counting.
  • Abstract. At this stage, the teacher models the mathematics concept at a symbolic level, using only numbers, notation, and mathematical symbols to represent the number of circles or groups of circles. The teacher uses operation symbols (+, –, ) to indicate addition, multiplication, or division.
On ways to do this:

CRA/CPA techniques and tools for Mathematics learning. (Source: The Access Center)
At secondary level, I would take the above to the next level, using the free for download GeoGebra application. As an illustration of its power, here is a video on studying the parabola:

(And for Physics, this study of the thin lens in action is very illuminating.)

We thus see how the pictorial and the abstract can be brought together, until hopefully some sparks catch.

As a bonus, we are seeing a case in point on how computer assisted instruction can be very powerful indeed. I am pretty well sure that after one session with this software, or even just watching the vid, the basis for the parabola will be understood by most students.

Let's just say, there is a reason for the saying "I see . . . " as a synonym for "I understand."

But what about other areas?

Obviously, pictures and diagrams help a lot. 

I am fond of the use of block diagrams that allow us to see how inputs, interactions and different facets lead up to an output behaviour. (In Control Systems, there is even a block diagram algebra!)

From such, we can then move to quantified models and use spreadsheets or the like to express resulting mathematical relationships. Where techniques like Laplace transforms can be used to reduce lumped dynamic effects to algebra. And of course calculus is the core technique that studies rates and accumulations of change, stocks and flows etc.
The essence of Calculus. To move to the next step, think about turning on and off the flow r(t), so that it follows a bell-shaped pulse. We can then slice that up into many step-like rectangles, width dt, d being a prefix for "so small that any smaller and it vanishes" -- infinitesimal.  Incrementally adding up areas under r vs t then gives us the increments of volume, that would follow a sort of S-shaped curve, the sigmoid, where growth in V vs t is at first slow, then speeds up as r vs t rises, then slows back down to a plateau as r falls back to zero as we gradually turn off the flow at the tail of the bell shaped pulse. To identify the slope of V vs t, we can imagine taking a chord from one point V0 to the next along V(t), V1and then allowing the second point to creep in towards the first point. Gradually the chord will tend to the tangent at the initial point V0. This is its limit, a major concept in calculus and other areas of Math. A plot of the slope of V vs t, dV/dt, will be r(t). Thus, we are at the heart of the Calculus, rates and accumulations, via areas under and slopes of. Where also we see that the "get the slope of" and the " get the area under" operations are mutually inverse. The find the slope operation is called differentiation, and is expressed as dV/dt. The get the area under operation is integration, and is symbolised by an elongated s called the integral sign.  Cf. here for a "made easy" from 100 years ago, & here for a modern free textbook from Wikibooks.
 The same power of visualisation extends even to subject like theology, for instance, here is a way that some have sought to understand the central theological mystery of the Christian Faith, the Triune God:

Symbols historically used to explain the triune God, where the Shield of Faith was actually seen as the Coat of Arms of God. The story is that the shamrock was used by Patrick in answer to the challenge of explaining the triune concept of God, as in is this one leaf or three? If one why three lobes, if three why one stem? And so if that mystery of tri-unity is hard to understand, what more can be said of the profounder one of God? The ICTHUS of course is a Greek MNEMONIC, an acrostic for Jesus Christ, Son of God, Saviour. The Triquetra extends the icthus to bring together the concept of complex unity that is at the heart of the trinity. And of course, complex unity, thanks to quantum physics, is now at the heart of understanding even matter: electrons etc behaving like particles or like waves depending on circumstances! Wave-particle duality.
We can thus see that a due emphasis on the concrete extended to the visual and onward to the abstract concept, is useful and powerful and widespread.

So, why not, let's try? END