Undergraduate students’ performance and confidence in procedural and conceptual mathematics

The general perception is that high school teaching of mathematics tends to be fairly procedural in South Africa and that students that enter university are better equipped to deal with procedural problems rather than conceptual. In this study we compare the conceptual and procedural skills of first year calculus students in life sciences. We also investigate students’ confidence in handling conceptual and procedural problems. The study seems to indicate that these students do not perform better in procedural problems than in conceptual problems. They are also more confident of their ability to handle conceptual problems than to handle procedural problems. Furthermore the study seems to indicate that, agreeing with common opinion amongst university teachers, students do not have more misconceptions about procedural mathematics than about conceptual issues. Introduction Mathematics pedagogy based on Vygotskian theory approaches mathematics as a conceptual system rather than a collection of discrete procedures. Vygotsky (1986) noted that the possibilities of genuine education depend both on the knowledge and experience already existing within the student (level of development) as well as on the student’s potential to learn. While this conceptual base is important, the communication system for algebra and other high school mathematics topics can be considered as a procedural approach to teaching mathematics. In South Africa, high school teaching of mathematics tends to be fairly procedural. Although most students arriving at university have well developed manipulation skills in mathematics, few of these students have really been exposed to deeper conceptual thinking. University teachers often complain that first year students have little understanding of any of the basic concepts of precalculus and even the better students are only better in a procedural way of thinking. With this background, we investigate the procedural and conceptual skills of first year students, also attempting to measure their confidence in handling procedural and conceptual approaches in first year calculus. Conceptual and procedural knowledge in mathematics and confidence of response Conceptual and procedural knowledge in mathematics is a topic addressed by many researchers. Most authors agree in principle but there are subtle differences in interpretation that we attempt to explicate in the following literature exposition. In its “Learning standard for mathematics” the New York State Education Department (2005) distinguishes between conceptual understanding and procedural fluency: Conceptual understanding consists of those relationships constructed internally and connected to already existing ideas. It involves the understanding of mathematical ideas and procedures and includes the knowledge of basic arithmetic facts. Students use conceptual understanding of mathematics when they identify and apply principles, know and apply facts and definitions, and compare and contrast related concepts. Procedural fluency is the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. It includes, but is not limited to, algorithms (the step-by-step routines needed to perform arithmetic operations). Hiebert and Lefevre (1986) describe procedural knowledge as "composed of the formal language, or symbol representation system ...[and] the algorithms, or rules, for completing mathematical tasks". They continue to assert that procedural knowledge is meaningful only if it is linked to a conceptual base. Teaching procedures and concepts Teaching for procedural knowledge means teaching definitions, symbols, and isolated skills in an expository manner without first focusing on building deep, connected meaning to support those concepts (Skemp, 1987). Teaching for conceptual understanding, on the other hand, begins with posing problems that require students to reason flexibly. Through the solution process, students make connections to what they already know, thus allowing them to extend their prior knowledge and transfer it to new situations (National Council of Teachers of Mathematics, 2000). Brown et al (2002) are of the opinion that conceptual knowledge means that students must make sense of mathematics. The difference between teaching for conceptual knowledge and the traditional way of ending chapters with application problems is fundamental. For conceptual learning the applications are often at the beginning of the chapter and the mathematics is drawn from them. They are not merely a place for applying previously mastered skills, time permitting, as is the case in a more traditional setting. So while they agree that both procedural and conceptual knowledge are important, the key issue, as they see it, is in the manner and order in which procedures and concepts are taught. “Teaching first for conceptual knowledge leads to the acquisition of procedural knowledge, but the converse is not true.” (Brown et al, 2002). Relationship between procedural and conceptual thinking. The main reason for the reform movement in the teaching of calculus courses (in particular) was the emphasis on procedural knowledge and it has been claimed (Haapasalo and Kadijevich, 2000) that students’ conceptual knowledge will necessarily increase their procedural proficiency. In Anderson’s model of learning (Anderson, 1995), learning begins with actions on existing conceptual knowledge. The student begins to internalise the procedures involved, leaving aside the conceptual knowledge from which the procedures arose. So the conceptual knowledge changes into procedural knowledge. The process of acquiring procedural knowledge depends upon existing conceptual knowledge and the knowledge gained by the repeated use of procedures (Byrnes and Wasik, 1991). For Piaget this process develops further. After the student has gained proficiency in procedural knowledge, a process of reflection begins and as a result new conceptual knowledge develops (Byrnes and Wasik, 1991). For Piaget conceptual knowledge and procedural knowledge are both integral parts of a single cognitive schema, they are not separate. In the Piaget model procedural efficiency is a requirement for metacognition and conceptual thought (Baker and Czarnocha, 2002). For Vygotsky (Vygotsky, 1986), on the other hand, algebraic thought begins with conscious reflection upon existing unconscious conceptual knowledge. In a study by Baker and Czarnocha (2002), they found that conceptual thought is independent of an individual's ability to apply his or her procedural knowledge, supporting Vygotsky’s view that development can proceed through reflection upon existing conceptual knowledge independently of the reflection due to repeated actions. A relevant question is whether it is possible to have conceptual knowledge/understanding about something without having procedural knowledge. Baker et al (2004) address this issue: “In [the] ‘traditional’ curriculum, concept development is viewed as arising from computational proficiency with relevant procedures.” Calculus courses often begin with a brief review of definitions and then focus on computational modeling of procedural knowledge grounded in algebra. Aspinwell and Miller (1997) agree with this view: “students regard computation as the essential outcome of calculus and thus end their study of calculus with little conceptual understanding.” The “dynamic action view” model (Haapasalo and Kadijevich, 2000) is an example of a model based on procedural knowledge. Here learning takes place by applying procedural knowledge to an existing conceptual foundation in which case increasing proficiency in procedural knowledge assists in expanding conceptual knowledge. Gray and Tall (1991, 1994) and Tall et al (2001) take the analysis to a next level. They introduce the idea of a procept as the symbolisation of an object that arises from processes carried out on other objects. Such procepts can then be viewed in two distinct but related ways, as a process or as an object. Procept is considered as a cognitive construct, in which the symbol can switch from a focus on process to compute or manipulate, to a concept that may be thought of as an entity that can be manipulated. Tall et al (2001) believe that procepts are at the root of human ability to manipulate mathematical ideas in arithmetic, algebra and other theories involving manipulable symbols. They allow the biological brain to switch effortlessly from doing a process to thinking about a concept in a minimal way. Tall et al (2001) consider the word procedure as a specific sequence of steps carried out a step at a time, while the term process is used in a more general sense to include any number of procedures with “the same effect.” Those who are procedurally oriented are limited to a particular procedure, with attention focused on the steps themselves, whilst those who see symbolism as process or concept have a more efficient use of cognitive processing. (Tall et al, 2001). Recognising the same distinction of components, Dubinsky et al (Dubinsky, 1991; Thomas and Holton, 2003) introduce the idea of processes being encapsulated as objects. They embed this encapsulation in what they call the APOS model for the construction of conceptual mathematical knowledge, describing how Actions become Processes that can be viewed as Objects, as part of Schemas. The components of this model: Action, Process, Object and Schema represent an increasing level of learning. An action is a change that an individual makes in a mathematical context requiring precise instructions to perform. A process takes place when the individual begins to have control over the concept. An object is constructed from a process when the individual becomes aware of the entire concept and understands that actions or processes can act on the concept. A schema is when objects and processes from more than one area can be combined in more than one way. Confidence of response The Confidence of Response Index (CRI) has its origin in the social sciences, where it is used particularly in surveys and where a respondent is requested to provide the degree of certainty he has in his own ability to select and utilise well-established knowledge, concepts and laws to arrive at an answer (Webb, 1994). In an academic examination environment a student is asked to provide an indication of confidence of response along with each answer set. This indication is usually based on some scale, say (0 – 5), where 0 implies a total guess and 5 implies complete confidence. Irrespective of whether the answer is correct or not, a low confidence indicates a guess which, in turn, implies a lack of knowledge. However, if the confidence is high and the answer wrong it points to a misplaced confidence in his knowledge on the subject matter, either misjudging his own ability or a sign of the existence of misconceptions. Hasan et al (1999) use the confidence of response, in conjunction with the correctness or not of a response can thus be used to distinguish between a lack of knowledge (wrong answer and low confidence) and a misconception (wrong answer and high confidence). This may not always be the case; students could just be over confident or in procedural problems students with high confidence may make numerical errors. Research question In view of the discussion above on procedural and conceptual knowledge in mathematics, we use the following working definitions for the two concepts for the purpose of this study: Procedural knowledge is the ability to physically solve a problem through the manipulation of mathematical skills, such as procedures, rules, formulae, algorithms and symbols used in mathematics. Conceptual knowledge is the ability to show understanding of mathematical concepts by being able to interpret and apply them correctly to a variety of situations as well as the ability to translate these concepts between verbal statements and their equivalent mathematical expressions. It is a connected network in which linking relationships is as prominent as the separate bits of information. The objective with the study is to determine whether there is any relation between students’ conceptual and/or procedural understanding in mathematics and whether there is any relation between their confidence levels when handling procedural and conceptual problems. Furthermore we want to investigate the relationships between students’ confidence and their actual performance in procedural and conceptual mathematical problems. The experiment In order to compare the conceptual and procedural abilities of students, we used a group of first year students in life sciences. These students all do an introductory course in applied calculus in the mathematics department. The number of students in the sample group is 235. The test consists of ten multiple-choice items of which five are considered to be predominantly procedural and five predominantly conceptual (by the authors). We realised that virtually all test items required both conceptual and procedural thinking to be solved satisfactorily. The items were mixed in the question paper. For construct validity, the test was thoroughly and independently scrutinised by colleagues in the mathematics department, all involved in the same course, to get an unbiased view of the the percentage of procedural knowledge (kp) and the percentage of conceptual knowledge (kc) needed to complete each item successfully according to the working definitions of the constructs mentioned earlier. The panel was also asked to indicate (in their opinion), the level of difficulty on a procedural (dp) and conceptual level (dc) of each item in the test. We used the averages of the panel members’ opinions after a discussion to confirm or dispute disparities in individual opinions. We realise, in agreement with Anderson (1995), that the concepts procedural and conceptual are not absolute – conceptual problems can become procedural if students are exposed to the same type of problem repeatedly. Despite this, an impressive cohesion of opinions was experienced. An example of an item considered to be (more) conceptual by the panel, is: Which of the following graphs f(x) satisfies both the given conditions (i) f ′(x) > 0 on (∞, 0) and f ′(x) < 0 elsewhere (ii) f ′′(x) > 0 on (∞, -1) and f ′′(x) < 0 elsewhere According to the panel, for this item: kp = 0.18, kc = 0.82 , dp = 1.25 and dc = 2.5. An example of an item considered to be more procedural by the panel, is: For the function f(x) = x3, what is the equation of the tangent line at x = 1? C D A B E. None of the above. A. y = x + 2 B. y = 3x – 2 C. y = x D. y = 3x + 2 E. None of the above For this item: kp = 0.89, kc = 0.11, dp = 1 and dc = 1 (according to the panel). On completion of each of the items, the student had to indicate how confident he/she was about the answer to the question, choosing between completely certain, fairly certain, not certain and a total guess. For each student we calculate a procedural performance index (PPI) and a conceptual performance index (CPI). The PPI is calculated using the formula