TheMathematics Educator 2009, Vol. 19, No. 1, 18–26 Understanding Rolle’s Theorem Revathy Parameswaran This paper reports on an experiment studying twelfth grade students’ understanding of Rolle’s Theorem. In particular, we study the influence of different concept images that students employ when solving reasoning tasks related to Rolle’s Theorem. We argue that students’ “container schema” and “motion schema” allow for rich concept images. without making heavy demands on their aptitude for Introduction abstract and rigorous mathematical understanding. Advanced mathematical concepts are characterized Some researchers argue that an introductory calculus by complex interactions between intuitive and rigorous course should be informal, intuitive, and conceptual, reasoning processes (Weber & Alcock, 2004). based mainly on graphs and functions (Koirala, 1997); Learning calculus, which involves processes pertaining formulas and rules should be carefully and intuitively to advanced mathematical thinking, has been a subject developed on the basis ofstudents’ previous work in of extensive research. One of the significant mathematics and other sciences (Heid, 1988; Orton conclusions arising out ofthis research is that students 1983). One of the guiding principles of teaching typically develop routine techniques and manipulative calculus could be the ‘Rule ofThree,’ (HughesHallet, skills rather than an understanding of theoretical et al., 1994) which says that, wheneverpossible, topics concepts (Berry & Nymann, 2003; Davis & Vinner, should be taught graphically, numerically, and 1986; Ervynck, 1981; Parameswaran, 2007; Robert, analytically. The aim is to balance all three of these 1982; Sierpinska, 1987). components to enable the students to view ideas from The subject ofcalculus is rich in abstraction and different standpoints and develop a holistic perspective calls for a high level of conceptual understanding, ofeach concept. where many students have difficulties. FerriniMundy There has been extensive research into the and Graham (1991)argue that students’understanding difficulties that students encounter in understanding of central concepts of calculus is ‘exceptionally limits, functions, differentiability, continuity, and so primitive’: on. However, there is not much literature on students’ Students demonstrate virtually no intuition about understanding of other concepts in calculus. Apart the concepts and processes of calculus. They from the cognitive obstacles that arise in the learning diligently mimic examples and crank out of calculus concepts due to the complexity of the homework problems that are predictably identical subject matter, students sometimes encounter to the examplesin the text. Misconceptionsexist difficulties inherent in mathematical reasoning. For as a result of student attempts to adapt prior example, deductive reasoning is a fundamental tool for knowledge to a new situation. Research suggests mathematical thinking; however, students reveal that students have a strong commitment to these serious difficulties developing such reasoning skills. misconceptions and that they are resistant to Orsega and Sorizio (2000) propose the mental model change and direct instruction (p. 631-632). theory of JohnsonLaird and Byrne as a cognitive While teaching calculus to a wide range of framework to analyze students’ difficulties in students, it is more practical to appeal to students’ deductive reasoning. Orsega and Sorizio argue that a intuition when conveying mathematical concepts and didactical model should be designed to enable ideas, building on what they have already learned undergraduates to overcome the fallacies of their deductive inferences. They consider a teaching Revathy Parameswaran received her PhD from Chennai Mathematical Institute. She has been teaching mathematics for method that enables first-yearundergraduates to make senior high students for the past eighteen years. Her research explicit the tautologically implicit properties in the interests include cognitive development and advanced hypothesis ofRolle’s Theorem and to reflect on them. mathematical thinking. Her hobbies include reading books and listening to classical music. 18 Understanding Rolle’s Theorem The purpose of this paper is to report on an changes its direction at some point to come down. experiment carried out to study twelfth grade students’ Rolle’s Theorem thus can be used to explain that the understanding ofRolle’s Theorem and its relationship velocity of the ball which is thrown upwards must to the closely related Mean Value Theorem. In become zero at some point (Berlinski, 1995). particular, I set up tasks designed to study (1) the Theoretical Background learner’s ability to state the theorem and apply it to reasoning tasks, (2)the influence ofconcept images in We shall use the concepts introduced by Weber his or her reasoning about the theorem, and (3) the and Alcock (2004) in their research on syntactic and learner’s ability to perceive the relationship between semantic proof production. A proof is referred to as Rolle’s Theorem and other related mathematical syntactic ifit involves only manipulation offacts and concepts. formal definitions in a logical manner without Forthe reader’s convenience, we recall below the appealing to intuitive and nonformal representations statement ofRolle’s Theorem. of the mathematical concepts involved. The prover need only to have the ability to make formal Let f be a function that satisfiesthe following three deductions based on the relevant definitions and hypotheses: (1) f is continuous on the closed interval [a,b], (2) f is differentiable on the open concepts. Such knowledge and understanding is called interval (a,b), and (3) f(a) = f(b). Then there is a syntactic knowledge or formal understanding. On the number c in (a,b) such that f'(c) = 0 (Stewart, otherhand, if the proveruses instantiations to guide the 1987). formal inferences, he orshe is said to possess semantic or effective intuitive understanding. Instantiation is Why Rolle’s Theorem? described as “a systematically repeatable way that an As observed by Berlinski (1995), “Rolle's Theorem individual thinks about a mathematical object, which is is about functions, and so a theorem about processes internally meaningful to that individual” (p. 210). represented by functions, an affirmation among other “Semantic or effective intuitive understanding is things about the coordination of time and space. ... The described as the ability on the part of the prover to constraints deal with the two fundamental explicitly describe how she could translate intuitive mathematical properties of continuity and observations based on instantiations into formal differentiability...” (191–192). Berlinski further mathematical arguments” (p. 229). While formal observes that: “Rolle's Theorem establishes a understanding is at the superficial level, effective connection between continuity and differentiability. intuitive understanding lies at a deeper level and is Continuity guarantees a maximum; differentiability characterized by the following features, which we delivers a number. Fermat's Theorem [which says that illustrate in the context ofRolle’s Theorem: iffhas a local extremum at c and iff'(c)exists, then • Instantiation “One should be able to instantiate f'(c) = 0] supplies the connection between concepts” relevant mathematical objects” (p. 229). Forexample, (196). The statement ofthe theorem involves multiple in the case of understanding Rolle’s Theorem, the hypotheses, the universal quantifier (for all) and the learner instantiates the statement of the theorem if he existential quantifier (there exists). Also Rolle's recognizes its applicability (ornonapplicability) in the Theorem offers the opportunity for pictorial, intuitive, case of a “typical” function -- not merely that of a and logical interpretations. quadratic-- possibly in terms ofa graph. The knowledge components required for the • Richness “These instantiations should be rich understanding of this theorem involve limits, enough that they suggest inferences that one can draw” continuity, and differentiability. The proof of the (p. 229). theorem is given using the Fermat’s Theorem and the • Accuracy “These instantiations should be Extreme Value Theorem, which says that any real accurate reflections of the objects and concepts that valued continuous function on a closed interval attains they represent” (p. 229). In ourcontext, the examples its maximum and minimum values. The proof of should not be too special as to suggest properties not Fermat's Theorem is given in the course while that of implied by the theorem. For example, one may be Extreme Value Theorem is taken as shared (Stewart, misled to believe that there is a unique point where the 1987). Hence we appeal to the learners'intuition rather derivative vanishes ifone always instantiates the graph than be rigorous in ourapproach. to be a parabola. The beauty ofthis theorem also reveals itselfin its •Relation to formal definition “One should be able connection with real life. A ball, when thrown up, to connect the formal definition ofthe concept to the comes down and during the course ofits movement, it instantiation with which they reason” (p. 229). Revathy Parameswaran 19 In theirfoundational work, Vinnerand Tall (1981) a period of six months, forming part of a yearlong have provided a framework for analyzing how one twelfth grade curriculum. Rolle’s Theorem was part of understands and uses a mathematical definition. the curriculum.