© 1996: The University of Newcastle: Faculty of Education
A structured solution format
A possible structured solution format might be presented in the following stages:
(1) The problem
In your own words, clearly describe the mathematical situation: what is the question asking?
(2) The plan
Please explain carefully how you think you will attempt the solution for this problem. (NOTE: the mathematical OBJECTS that the problem presents and the ACTIONS suggested by each of these).
(3) A solution
Now describe each step of your solution.
(4) Evaluation
What steps are you taking to CHECK your results? What other steps COULD you take?
This four-step model is based upon a synthesis of that used for Newman Analysis procedures (1983) and upon a cyclic model of mathematical interaction developed by the author. The Newman model has been considered elsewhere; the cyclic interaction model is outlined here in more detail.
A recent study (Arnold, 1996) examined the ways in which individuals (school students and preservice teachers) used a range of available mathematical software tools for the learning of mathematics. The periods of interaction ranged for the students from six months to two years, on a weekly basis involving hour-long individual tutorial situations utilising a computer-based learning environment created for this purpose (Arnold, 1993). Although the action occurred within the domain of algebra and involved the use of computer software tools, the findings offer useful information regarding the ways in which individuals engage in problem situations within a mathematical learning environment based upon the principles of challenge and support.
Figure 3 illustrates the process by which individuals engage in mathematical interaction. Confronted by a problem situation (which may be formal or informal, explicit, as in a "textbook question," or implied, as in a word problem), the student must interpret the situation and recognise a mathematical object, which may be an equation, graph, symbol or perhaps a geometric figure. Failure to recognise such an object within a particular mathematical situation may not mean that no further mathematical actions can be effected. It does, however, negate the possibility of software tool use within that context, since such use requires a specific object upon which to act.
Figure 3: A model of mathematical interaction
This object signals mathematical action, the strength of the signal character (van Hiele, 1986, pp. 60-61) influencing the available repertoire of strategies (which may include traditional mathematical actions, such as simplifying, substituting, solving, expanding, factorising, sketching, and computer-based actions, such as graphing, tabulating or even animating). Certain forms displayed strong and consistent signal characters which readily led to action on the part of participants. Most notable of these were two forms of equations encountered: y = 2x - 1 and 2x - 1 = x + 7. The former was invariably associated with graphing, and the majority of participants demonstrated the ability to deduce useful graphical meanings (most particularly gradient and y-intercept information) from this algebraic form. The second induced in all participants an automatic (and at times overwhelming) action sequence, leading to the production of a "solution." Graphs, too, (both linear and quadratic) presented familiar and meaningful information to most participants.
While some forms displayed strong signal character, simple algebraic expressions (such as 4 - 3x) induced some frustration and confusion among even the more experienced participants. All expressed a desire to act in some way upon the expression, but were unable to do so with their existing repertoire of available mathematical actions. The expression (x - 1)(x + 1) was not associated with the same reactions, since it permitted a familiar mathematical action (expansion), and so induced a sense of closure.
The action strategy produces a result which must be evaluated by the student regarding the extent to which it brings the encounter closer to a point of closure. Available computer tools were used most commonly at this point to verify results obtained by traditional means. Again, the metaphor of the "answer in the back of the book" appears to apply well here.
Traditional mathematical actions as observed in this study tended to move relatively quickly to a point of closure. Algebra was commonly associated with obtaining an "answer," usually through the application of a well-defined sequence of steps. Such a perception appeared largely incompatible with a focus within the learning environment upon open-ended exploration. In fact, the readiness with which even high ability students would conclude their computations, even while expressing less than full confidence in their results, was noted as a source of some concern. When limited to traditional methods, then, the stage of evaluation appears likely to conclude the mathematical process with brief verification using whatever means are available.
Use of computer tools, however, appeared to encourage a cyclic aspect in this process. Even when used only for verification of results through use of an alternative representation (such as viewing a graph to check the solution of an equation), the user is presented with what is effectively a new mathematical object or situation, requiring further interpretation and the possibility of subsequent action. It is, of course, a very small step, too, from verification of results to verification of conjectures, a far more active and appropriate mathematical approach.
Applying Theory to Practise.
A graduated help facility.
Last updated: 1st May, 1996
Stephen Arnold
crsma@cc.newcastle.edu.au
© 1996 The University of Newcastle
© 1996: The University of Newcastle: Faculty of Education