Thursday, August 23, 2012

Capacity focus, 55e: Remedial -- and perhaps first time through -- Mathematics in light of the Knowledge Space Theory concept of Jean-Paul Doignon and Jean-Claude Falmagne (A second possible application of the spiral, individualised window of learning opportunity approach)

(Two Sigma/Digital learning transformation series 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 )

 With Tropical Storm Isaac moving on after a side-swipe, let's return to the question of renewing our approach to education. In this case, for Mathematics, this is under the shadow of shocking regional CXC results this year [looks like about 33% overall passes . . . ], which (HT: reader "X") have excited sharp comments in several quarters.

To begin, let us first extend the Vygotsky Zone of Proximal Development -- window of learning opportunity -- concept. We can do this, based on aspects of the pioneering work of Jean-Paul Doignon and Jean-Claude Falmagne who in the mid 1980's introduced the concept of Knowledge Spaces and the linked idea of the knowledge state of a given student identifiable though the concepts and skills he has mastered and/or is ready to now master by building on what s/he already has learned.

For instance:

Or, in more details of the algebra section from a knowledge space map (of 397 problem types) of Arithmetic, and Algebra up to pre-calculus level, where (a) and (b) are from the same table:

{U/d Aug 24} This can be set in the wider context pioneered by Singapore, i.e the Pentagonal Mathematics Knowledge, Attitudes & Skills Framework:

The overall knowledge space (which is a requisite for identifying a learner's knowledge state at any given time, thus also to monitor progress) can be compiled by various techniques. Wikipedia summarises usefully:
there exist several methods to construct knowledge spaces. The most frequently used method is querying experts. There exist several querying algorithms that allow one or several experts to construct a knowledge space by answering a sequence of simple questions.[6][7][8]
Another method is to construct the knowledge space by explorative data analysis (for example by Item tree analysis) from data.[9][10] A third method is to derive the knowledge space from an analysis of the problem solving processes in the corresponding domain.[11]
 These techniques can get highly mathematical and quite complex (with very interesting bits of applied set theory and even more fascinating possibilities for algorithms), of course. 

But such complexity -- through clever programming and attractive interfaces -- can be hidden behind the interface to a computer application suite that can identify the knowledge state of a given learner, allowing for individualised curricula. (This can obviously be extended to in effect any subject, but it probably makes sense to focus on the "core of cores" in the curriculum where our region is struggling, Math and English.)

For example, we may look at the ALEKS commercial application. (Video preview here. Note tour here and available courses here.) 

I have found a teacher's guide to ALEKS here at Youtube:

(NB: Do, forgive the fuzzy screen-cast. However, the branching or "case" structure slide show with interactions/activities approach should be clear enough, and this is right up XERTE's street. It is also obvious that the sort of screen-capture explanations that the Khan Academy specialises in or video demonstrations or even exercises with teacher-developed "shoebox kits" of hands-on experimental objects with instructions could easily be incorporated (or of course, various Math kits -- I am partial to the Calvert School's store resources), making for a very rich, stimulating and interactive, individualised learning experience. There is plainly a very large base of pre-loaded problems in the program, and maybe a problem-generating engine. BTW, there is a special module for getting what Americans call "Math Facts" -- addition/subtraction and multiplication/ division solidly learned. I suspect, that is one underlying problem. I like the visualisations provided by Autograph for Maths, from the UK.  Boardworks (also from the UK) provides some very useful resources keyed to interactive classroom whiteboards [not as "hot" as the Autograph ones but quite useful], and covers a wide range of GCE type syllabi at O and A levels, but is a bit expensive.)

Such an application could of course be integrated with the same kind of multimedia seminar room previously discussed:

ALEKS, of course, is keyed to the American educational system, so it is not directly relevant to CXC-type syllabi. And, while the CXC syllabi are not available online (apart from, by purchase of a print copy), the approach is close enough to that of the UK-based Cambridge GCE Math D syllabus 4024 (June 2014) -- notice, one paper is no-calculator, the other permits calculators, a "scientific & trigs calculator" being strongly suggested -- that we can draw close parallels:

1: Number
2: Set language and notation
3: function notation
4: squares, cubes and matching roots
5: Directed numbers (+/- values)
6: Vulgar and decimal fractions and percentages
7: Ordering by magnitude and relationships:  =, ≠, >, <, 
8: Standard (scientific) notation
9: The four basic arithmetic operations with precedents (BOMDAS)
10: Estimation and rounding
11: Limits of accuracy
12: Ratios, proportion, rate
13: Percentages
14: Use of a calculator
15: Measures
16: Time on 12 and 24 hr clocks
17: Money, including conversion
18: Personal and household finance, including tables and charts, profit/loss, simple interest
19: Graphs in practical situations (including kinematics of speed & distance)
20: Graphs of functions
 21: Straight line graphs, y = mx + c etc
22: Algebraic representation and formulae
23: algebraic manipulation
24: Laws of indices
25: Solving equations and inequalities (Linear, fractional and quadratic)
26: Graphing inequalities (Linear programming NOT included)
27: Basic geometrical terms, figures & relationships
28: Geometrical constructions
29: Bearings from 000 to 360 degrees
30: Symmetry
31: Angles
32: Loci
33: Mensuration (esp. for basic figures)
34: Trigonometry (not solution of identities)
35: Basic statistics (Not inferential)
36: Basic probability
37: Basic matrices (inversion is of 2 x 2)
38: Transformations in the XY plane (2 x 2 matrices)
39: 2-D vectors (but not extensions to complex nos!)
This isn't Grandpa's maths!

The simple listing of topics shows how complex and hierarchical the modern type of Math syllabus is, and how demanding it is of understanding. 

For instance, you cannot apply matrix transformations without understanding simultaneous linear equations and how one gets to a matrix and matrix operations from such. In addition, one would need understanding of how a matrix applied to a vector transforms it into a different one, thus leading to transformation of a figure in the plane. (This is connected to Computer Graphics. BTW, extending to three dimensions, a 3-d rotation matrix allows us to deduce the look angle for pointing an antenna at a satellite in geostationary orbit 23,000 miles up, over a point on the Equator. As an be imagined, many transformations used in computer graphics are based on matrices. Matrices are also intimately involved in many aspects of science and engineering, e.g. in modelling circuits and -- extending to mathematical operations and transforms, in the state space approach to control systems.)

 The low performance of many students is thus not hard to understand. 

Moreover, given the hierarchical pattern of the knowledge space with ever so many dependencies from one capacity to the next and onward, once a child falls behind, it is easy for the child to be run over, and left ever further behind, lying and bleeding in the rear view mirror.

We have to break this cycle, and worse this needs to be in a situation where we have to recognise -- per Piaget and others -- that not all students mature to a level of mental capacity that they can handle abstract mental concepts and formal operations at the same general time. The point that this may well be in part biologically linked, similar to puberty (which obviously happens at different ages for different children) then introduces a justice issue. 

Yes, a JUSTICE issue.

Mathematics capacity is not only critical to access many of the professions that are critical to national and regional progress, but also to access careers that have high upward social mobility.

So, it is simply not good enough to use correctable gaps in Math knowledge as socially loaded filters that lock out people who will disproportionately come from and be relegated to lower social strata, and of course, girls.  

That is, as a matter of justice, we need to think out ways to help our children master and credibly document mastery of a critical core of mathematics for development, especially at secondary level  -- including second chance secondary level . . . that flunked out 60 - 70% should not be simply written off -- and to open gateways for extending the "mastery of Math" knowledge base across time as people who were hampered by their previous level of development gain enough capacity to master abstract concepts and skills. 

I suggest, first, that this is possible on a modular stage by stage spiral basis that allows for individualisation of learning paths and styles based on the power of modern digital technology:

Clearly, units of instruction should have a heavy diagnostic component, and should then address individual cases through targetted skills building. The knowledge space and knowledge state concepts above are perfect for this. 

Also, the approach offers an obvious shift: FROM grading relative competence on a standard one-size-fits-all syllabus, TO a cluster of learning modules, where what is developed, assessed and put in the portfolio of learning is a growing list of demonstrated, mastered content and skills. That way, students move away from a one-point grade assessment to a profile of competencies that are linked to important areas of achievement.

Computer technology allows this to be done, and thresholds can be identified and listed as a sequence of grades of achievement relevant to onward studies or job requisites.

Where also, we need to remind ourselves of the statistics that say that even in advanced societies, only about 30 - 35% of adults are fully capable of abstract operations:

There is a body of evidence that even in advanced countries only about 1/3 of adults achieve abstract operations capability

It is helpful to give an idea of what the identified Formal Operations are about. Saul McLeod of Simply Psychology gives a good clip:

The formal operational stage begins at about age 11. As adolescents enter this stage, they gain the ability to think in an abstract manner, the ability to combine and classify items in a more sophisticated way, and the capacity for higher-order reasoning.

At about age 11+ years, the child begins to manipulate ideas in its head, without any dependence on concrete manipulation; it has entered the formal operational stage. It can do mathematical calculations, think creatively, use abstract reasoning, and imagine the outcome of particular actions.

An example of the distinction between concrete and formal operational stages is the answer to the question “If Kelly is taller than Ali and Ali is taller than Jo, who is tallest?”  This is an example of inferential reasoning, which is the ability to think about things which the child has not actually experienced and to draw conclusions from its thinking.  The child who needs to draw a picture or use objects is still in the concrete operational stage, whereas children who can reason the answer in their heads are using formal operational thinking.
For instance, a child in the formal operations stage can "easily" design a systematically structured simple experiment (such as to explore the parameters that govern the back-forth swinging of weights suspended from a hook using a string of adjustable length -- pendulums) without detailed step by step concrete instructions. That is because such a child can think abstractly about scientific methods, laws, possible consequences, and the like. But, without step by step "scaffolding," probably detailed sketches and maybe a live demonstration, a child not yet at that level will be likely to flounder.

Indeed, I suspect that for many children in our region, doing math boils down to learning by concrete example how to do a Type X1 problem, then an X2 and an X3 etc. Throw in a similar Type Y that if the underlying principle is understood it can be solved easily enough (but is not directly comparable to the Type X's), and they will flounder and likely get stuck.  And maybe that is where Skemp's contrast of instrumental and relational understanding as raised earlier in this series comes in:
It was brought to my attention  some years ago by Stieg Mellin-Olsen, of Bergen University, that there are in current  use  two  meanings  of  this  word.  These  he  distinguishes  by calling  them  ‘relational  understanding’  and  ‘instrumental  understand-ing’. By the former is meant what I have always meant by  understand-ing, and probably most readers of this article: knowing both what to do and why.  Instrumental understanding I  would  until  recently  not  have regarded as understanding at all. It is what I have in the past  described as ‘rules without  reasons’, without  realising that  for many pupils  and their teachers the possession  of such a rule, and ability to use it,  was why they meant by ‘understanding’ . . .
 So even words like "understanding" may have pitfalls in them.

However, it is worth following McLeod to the next step on possibilities for helping those who are ready make the leap:
Robert Siegler (1979) gave children a balance beam task in which some discs were placed either side of the center of balance. The researcher changed the number of discs or moved them along the beam, each time asking the child to predict which way the balance would go.

He studied the answers given by children from five years upwards, concluding that they apply rules which develop in the same sequence as, and thus reflect, Piaget's findings. Like Piaget, he found that eventually the children were able to take into account the interaction between the weight of the discs and the distance from the center, and so successfully predict balance. However, this did not happen until participants were between 13 and 17 years of age. He concluded that children's cognitive development is based on acquiring and using rules in increasingly more complex situations, rather than in stages.
 Since we have things like autistic savants that can do astonishing things in a few areas but are otherwise often severely retarded and even very capable and "bright" people have gaps in capability, perhaps, we need to adjust. Perhaps, there is a progress in stages and in particular areas, and as we reach a critical mass, there is a transformational "jump," an aha reaction. Where, a very encouraging and stimulative environment encourages not just assimilating new experiences to old conceptual and operational schemas in our minds, but accommodating to new experiences, promoting transformation of our thinking. Where also hands-on, minds-on activities/exercises and rich visual and verbal stimulation as well as talked- and- walked- through examples led by people who truly understand what they are doing themselves, can obviously make a big difference.

Certainly, for example: for me music is a mystery, one that I deal with in a very concrete way. 

I can understand the physics and the system of harmonics [I keep wanting to pull out an oscilloscope and plotting waveforms and Fast Fourier patterns then 3-d waterfall plots . . . ], but don't ask me to understand the stuff of artistic composition and the difference between styles, much less the technical terms that apply. I suppose, with a major effort and the sort of supportive scaffolding I could learn, but I suspect my time would be better applied elsewhere if this is going to require a major sustained effort to study.

But, if someone could figure out a way to turn learning music into an entertaining and interesting game . . .  

(Hence, the question of the power of entertainment in education, for good or ill. But also, we have to develop the discipline that sees that if something is important enough, "that's boring . . . " is not good enough to walk away or refuse to put in the requisite effort. Learning is often an acquired taste, and we need strong motivation to make the effort and bear with inevitable drudgery and frustrations. Oh, how I remember working through dozens and dozens of math problems per week, set by a Math teacher who believed in homework and "the Hobartian method of self-correction." That is, we were expected to work through solutions with the teacher -- and, often enough, selected students called on to report from their seats or sent to the board to show how they did the exercise -- and do the corrections for our own homework. Sometimes, if memory serves, we were then called up to have a look at our self-grading, one at a time.)

Back to the math issues . . . 
It seems to me that we may be trying to do too much for most children with one common "one size fits all" syllabus, and may be unfairly labelling other syllabi that do not so sharply depend on mastering a large cluster of heavily abstract operations before giving any certification of achievement, as "inferior."

As labels for children who are dismissed and dumped as failures coming out the starting gates, in short. 

It may, then, well be appropriate for us to develop a more modular approach to mathematical competence certification and provision to add in modules as one moves along. 

Surely, that could be more promising than doing an all or none lumped certificate, where also -- given the huge diversity in the topic outline as listed -- modules can be adapted to where children are heading, career-wise. And, maybe we could make provision for students to build up a profile of performance across time, adding further modules to a higher grade of certification?

(Perhaps, too, the modules could be keyed to a half-term length "standard" unit. That would help us do the plug-n-play game that is so helpful with planning and management. )

Why not distinguish numeracy and math, and focus numeracy on hands on, concrete object oriented math for the world of basic "practical" practice, where we work with concrete objects and related quantities that we can measure or observe directly? 

Then, perhaps a core numeracy could be covered up to about 3rd form, and thereafter we have diverse math programmes for different people, with modularity to fill in gaps as required?

But then, I must defer to those whose educational expertise is in Math, in the end.

Having said something about possibilities, we have to deal with the present reality for the time being. 

Remember, 2/3 of those just sent to sit math in our region "failed." After five years of secondary education.

That sort of "rejects" rate is intolerable.

As a first step, we need to move up to a better level of capacity to build understanding and competence, step by step. The first requisite for this would be accurate assessment of knowledge state in knowledge spaces on a regular basis and the creation of well thought through sequences of units of study, using software. (The ALEKS model could be a guide, but there is no reason why we cannot develop our own.)

This could serve in the first- time- through setting, but would obviously be more oriented to the remedial setting.

In this case, a knowledge state diagnosis would logically lead to an individualised sequence of units tailored to the needs of the learner, his or her readiness to handle abstract concepts and formal operations, and the like. One thought is that it may well be worth exploring the power of visualisations that help students to see what they are dealing with, and of course, experience with concrete and realistic cases will also help.

(My dad, in chatting with me about the upcoming storm, reminded me of a young man -- now a Math teacher -- who hated co-ordinates until he visited our home when there was a leak in the roof. By counting numbers of tiles out from the walls to give the location of the leak, we could tell the workman just where to patch. The light bulb went off as the young man realised this was a co-ordinate reference. He went on to major in Mathematics and to minor in education, and is now a Math teacher.)

So, let us see what is possible to begin a better approach to math education, first time through and remedial. More next time. God bless. END