Section (vi) The Novices' Difficulty with Grounding Intuitive Arguments on Appropriate Theorems. Decontextualised Knowledge, Regression to Quasi-Formal Familiar Modes of Reasoning and the Examples of the Intermediate Value Theorem and the Inverse Function Theorem
Context: See
Extract 7.6Structure: In the following, the tutor and the students discuss an application of the Intermediate Value Theorem and the Inverse Function Theorem. Both students have drawn the graphs in B7 and from them they have intuitively drawn conclusions. Implicitly they have used theorems which they cannot recall. Moreover, even when they reproduce the statement of a well-known theorem, they cannot link it to the intuition that led to answer the questions in B7.
The Episode: A Factual Account. See
Extract 7.6An Interpretive Account: The Analysis
In parts a and b of question B7, the students have drawn the graphs and given intuitive explanations of their (correct) answers. What they cannot do is support their arguments with formal explanations that go beyond the graphical representations of the functions in question and ground their arguments on theorems that they have been recently taught.
By convention, the questions in a weekly problem sheet usually aim at providing applications of the theorems proved in the lectures. Thus the theorems are seen into a context of applicability and are embedded into the growing domain of the novice's mathematical knowledge. Abidul and Frances do not seem to be in tune with these aims. Both employ A-level techniques (first and second derivative tests) to identify the local max and min of the function and use intuitive notions of bound and limit in order to construct the graphs of x5-5x+10 and x-1ex. So they seem to deviate from the enculturating contextualisation aimed at by the authors of the problem sheet which contains B7.
Characteristically when Abidul, under the pressure of the tutor who insists on Abidul grounding her illustrative, intuitive argument (A1) on a well-known theorem, replies hesitantly 'Rolle's', as if she feels obliged to hide behind some impressive name-dropping. Similarly Frances drops in IVT. Both students seem to drop the names at random, because they have heard them recently and they expect them to appear frequently. F1, A2 and A3 are fragmentary reconstructions of IVT. They are also evidence of how disconnected the students' actual mathematical thinking is from the 'official' mathematical knowledge that is supposed to form and support this mathematical thinking.
The situation repeats itself identically in part b of question B7. Again via the derivative tests, the min of the function has been identified and the graph has been drawn. Frances' argument about the smallness/bigness of x-1ex with relation to the size of x, is underlain by an intuitive and informal notion of limit and her answers are definitely embedded in the graph of x-1ex. Frances explicitly and exclusively turns to the graph in order to answer the question in the second part of b which requires a construction of a function by reversal.
That the second part of b is the most cognitively difficult task in question B7 is evident in Abidul's giving up and Frances' complete reliance on the graph. Frances carries out the reversal (which in thorough terms is allowed by the Inverse Function Theorem under certain conditions satisfied in question B7) required by the question by simply reflecting the decreasing part of the graph in the y axis and by locating the range of values of x that correspond to [e, +¥). This is exactly what the IFT allows but, as with the IVT, the students use the theorem unconsciously keeping thus its assumptions implicit. Both IVT and IFT — especially IVT — make quite obvious statements (if f is continuous on [a,b] then it takes all the values between f(a) and f(b); if f is continuously differentiable on an interval then its inverse is defined) that, however, hold under certain conditions. Novices tend to over-generalise the domain of validity of theorems only because the theorems apply to a large number of functions they are familiar with. Almost certainly it did not occur to Abidul and Frances that they needed to address a theorem to ground their answers to B7.
The difference between the tutor's and the student's approaches is of a delicate ontological nature. For the tutor IFT guarantees the existence of f-1 in B7 but for Frances (F2) this function already exists: it is there, on the graph, and in order to see it, all she has to do is look at the graph in a slightly new way (x becomes y and so domain becomes range). Frances does not need anything stronger than her own sight to be convinced and she assumes that everyone else shares her sense of conviction.
Conclusion: In the above, the students have been employing quasi-formal techniques (intuitive arguments based on graphical representations of functions) in order to
• find the roots of an equation,
• identify the image of a function, and
• construct the image of the inverse of a function.
Their arguments could have been grounded on the Intermediate Value Theorem and the Inverse Function Theorem but this did not occur to the students who, perpetuating A-level attitudes, used the theorems tacitly. The tutor has played the role of an agent trying to link the students' intuitions with the appropriate theorems and to foster in their minds the necessity of doing so. When they finally realise and put into practice the need to go beyond unconscious and unfounded uses of theorems, it will probably be a sign of adaptation to customary and formally acceptable mathematical practices.