Math Reasoning Calculus Textbook Exercises by Lithner

Posted on September 23, 2012 by


Lithner, Johan. 2004. “Mathematical Reasoning in Calculus Textbook Exercises.” The Journal of Mathematical Behavior 23 (4): 405–427. doi:10.1016/j.jmathb.2004.09.003.


  • However, in this paper I will: (i) study more carefully some of the mathematically superficial strategies and reasoning possible to use when solving exercises; (ii) propose a framework that can be applied to study students actual work with textbook exercises. (406)


  • the types of exercise reasoning that are encouraged by teachers and the types seen as suitable by the students. Teaching and learning can take place in many different ways, but among Swedish undergraduate students it is common to normally spend at least half of their study time working with textbook exercises (this assertion is based on local unpublished surveys and discussions with teachers). This reason to study textbooks is complemented by Love and Pimm (1996): “The book is still by far the most pervasive technology to be found in use in mathematics classrooms. Because it is ubiquitous, the textbook has profoundly shaped our notion of mathematics and how it might be taught. By its use of the ‘explanation–example–exercises’ format, by theway in which it address both teacher and learner, in its linear sequence, in its very conception of techniques, results and theorems, the textbook has dominated both the perceptions and the practices of school mathematics.” (406)
  • A series of studies (Bergqvist, Lithner, & Sumpter, 2003; Lithner, 2000, 2003b), trying to characterise students’ reasoning and argumentation in different situations, indicate that a main cause behind students’ learning difficulties is a focus on different superficial reasoning types. There are only rare occasions of mathematically well founded constructive reasoning, which can be summarised as: A version of the reasoning structure (1–4) is called plausible reasoning (abbreviated PR) if the argumentation:

    (i) is founded on intrinsic mathematical properties of the components involved in the reasoning, and (ii) is meant to guide towards what probably is the truth, without necessarily having to be complete or


  • There will be a distinction between intrinsic and surface mathematical properties of the components involved in the reasoning. An intrinsic property is deep and central to the component in a specific situation (see the analysis of Exercise 1.2.14 in Section 3.3 for an example). A surface property is not a key feature of the core part of a solution. When considering such a property it is not necessary to understand the central mathematical ideas and analyse the consequences of their properties (407)
  • There seem to be different superficial reasoning types commonly used by students, largely depending on the task solving situation. Some brief examples are:

    The keyword strategy in elementary arithmetics, e.g., subtracting if the exercise contains the keyword “less” (Hegarty, Mayer, & Monk, 1995; Schoenfeld, 1991).

    In (Bergqvist et al., 2003) upper secondary calculus students were found to apply repeated algorithmic reasoning (abbreviated RAR):

    1. (i)  The general strategy choice is to repeatedly apply algorithms, where each local strategy choice is founded on recalling that a certain algorithm will (probably) solve a certain task type. The algorithms are chosen from a set of (to the reasoner) available algorithms that are (to the reasoner) related to the task type by surface properties only.
    2. (ii)  The strategy implementation is carried through by following the algorithms. No verificative argu- mentation is required. If an implemented algorithm is stalled or does not lead to a (to the reasoner) reasonable conclusion, then the implementation is not evaluated but simply terminated and a new algorithm is chosen. (407)
  • The university students in Lithner (2000) used, to a large extent, reasoning based on established experiences (abbreviated EE), where the argumentation:
    (1) is founded on notions and procedures established on the basis of the individual’s previous experiences from the learning environment, and
    (2) is meant to guide towards what probably is the truth, without necessarily having to be complete or correct. (408)
  • A suitable framework for discussing this question is provided by Schoenfeld (1985, p. 15):
    “Resources: Mathematical knowledge possessed by the individual that can be brought to bear on the problem at hand. Intuitions and informal knowledge regarding the domain. Facts. Algorithmic pro- cedures. ‘Routine’ non-algorithmic procedures. Understandings (propositional knowledge) about the agreed-upon rules for working in the domain.
    Heuristics: Strategies and techniques for making progress on unfamiliar and non-standard prob- lems: rules of thumb for effective problem solving, including: Drawing figures; introducing suitable notation. Exploiting related problems. Reformulating problems; working backwards. Testing and verifications procedures.
    Control: Global decisions regarding the selection and implementation of resources and strategies. Planning. Monitoring and assessment. Decision-making. Conscious metacognitive acts.
    Belief Systems: One’s “mathematical world view”, the set of (not necessarily conscious) determi- nants of an individual’s behaviour. About self. About the environment. About the topic. About mathematics.” (424)
  • Schoenfeld (1991) has described that students are inclined to answer questions with suspension of sense making, and that they often use shortcut strategies. According to Doyle (1986, 1988), there is a pressure from students to reduce ambiguity and risk, and to improve classroom order, by reducing the academic demands in tasks. This may also be the case in textbook organisation and can be achieved by including a large proportion of IS and LPR exercises. Textbooks can be one of the sources that specify the academic demands, but Dreyfus (1999) argues that in textbooks students are rarely given explicit instructions or indications concerning the required quality of reasoning. In a historical perspective McGinty, VanBeynen, and Zalewski (1986) analysed grade 5 arithmetic textbooks from 1924, 1944, and 1984, and found that the number of word problems had decreased, the number of drill problems had increased, and that word problems had also become shorter and less rich. A brief comparison between some older calculus textbooks, for example, Courant and John (1965) and de La Valle ́e Poussin (1954), and the ones analysed in Section 4 indicates that the proportion of IS and LPR exercises have increased considerably. All this may be part of a self-deceptive way in the present mass education situation to continue, at the surface, to deal with advanced mathematical concepts in our calculus courses while perhaps seldom or never actually considering intrinsic properties. (425)

  • About 90% of the exercises are IS or LPR that can mainly be done by searching the text for methods. Therefore students may develop strategies where the question “what method should be applied?” is immediately asked, instead of first trying to reach a qualitative representation (Hegarty et al., 1995) of the task and base the solution attempt on the intrinsic mathematical properties of the components involved. (426)