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.
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.
Another method is to construct the knowledge space by explorative data analysis (for example by Item tree analysis) from data. A third method is to derive the knowledge space from an analysis of the problem solving processes in the corresponding domain.
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:
CAMBRIDGE GCE MATH D:
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
14: Use of a calculator
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
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!)
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|
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.
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.
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.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.
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.
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 . . .
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. )
But then, I must defer to those whose educational expertise is in Math, in the end.
Remember, 2/3 of those just sent to sit math in our region "failed." After five years of secondary education.