{"id":1238,"date":"2016-05-02T08:00:23","date_gmt":"2016-05-02T12:00:23","guid":{"rendered":"http:\/\/blogs.ams.org\/matheducation\/?p=1238"},"modified":"2016-05-01T13:31:49","modified_gmt":"2016-05-01T17:31:49","slug":"a-framework-for-integrating-conceptual-and-procedural-understanding-in-the-first-two-years-of-undergraduate-mathematics","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/matheducation\/2016\/05\/02\/a-framework-for-integrating-conceptual-and-procedural-understanding-in-the-first-two-years-of-undergraduate-mathematics\/","title":{"rendered":"A Framework for Integrating Conceptual and Procedural Understanding in the First Two Years of Undergraduate Mathematics"},"content":{"rendered":"

By\u00a0Karen Keene<\/a> and\u00a0<\/span>Nicholas Fortune<\/a>, North Carolina State University <\/span><\/em><\/p>\n

One common instructional approach during the first two years of undergraduate mathematics in courses such as calculus or differential equations is to teach primarily analytic techniques (procedures) to solve problems and find solutions. In differential equations, for example, this is true whether the course is strictly analytical or focuses on both analytic techniques and qualitative methods for analysis of solutions. <\/span><\/p>\n

While these analytic techniques play a major part of the early undergraduate mathematics curriculum, there is significant discussion and research about the importance of learning the concepts of mathematics. Many researchers in mathematics education encourage teaching mathematics where students learn the concepts before the procedures and are guided through the process of reinventing traditional procedures themselves (e.g., Heibert, 2013). Additionally, educators who have developed mathematical learning theories often set up a dichotomy between the two kinds of learning (e.g., Skemp, 1975; Haapasalo & Kadijevich, 2000). At the collegiate level, we as professors may agree that these educational ideas hold merit, but also firmly believe that students have a significant amount of content to learn and may not always be able to spend the time necessary to allow students to participate fully in the development of conceptual understanding and the reinvention of the mathematics (including procedures). <\/span><\/p>\n

However, some researchers, including ourselves, provide evidence that \u201cteaching the procedures to solve problems and find solutions\u201d and \u201cproviding ways for teaching concepts first so students will truly understand\u201d can be integrated, and that the notion of learning procedures does not need to be shallow and merely a memorized list (Star, 2005; Hassenbrank & Hodgson, 2007). Our framework to merge these two ways of teaching is titled the <\/span>Framework for Relational Understanding of Procedures<\/span><\/i>. It was developed as part of Rasmussen and colleagues\u2019 work in differential equations teaching and learning (Rasmussen et. al., 2006). Skemp coined the original definition; she defines relational understanding as \u201cknowing both what to do and why\u201d and contrasts it to instrumental understanding as \u201crules without reason\u201d (1976, p. 21). <\/span><\/p>\n

Following, we describe the six components of the <\/span>Framework for Relational Understanding of Procedures<\/span><\/i>. The idea is that each category can be used to consider and enhance students\u2019 learning as they study a procedure. \u00a0For each one, we provide a brief explanation, questions about student thinking, and an example of an exam question related to each component taken from our work in differential equations. \u00a0Likely, each instructor could add other algorithms in differential equations as well as other courses. <\/span><\/p>\n

<\/p>\n

Components of Relational Understanding of Procedures<\/b><\/p>\n

Student can anticipate the outcome of carrying out the procedure without actually having to do so and they can anticipate the relationship of the expected outcome to outcomes from other procedures. \u00a0<\/b><\/p>\n

This component suggests that a student understands what kind of solution would be expected before solving. A student might need to consider the following: Is the solution going to be a number, or a function? When is the solution one or two functions? Are there different forms to show the answer? How do the answers compare to other answers from similar procedures?<\/span><\/p>\n

Example:<\/span><\/p>\n

Suppose that a differential equation can be solved with either separation of variables or with a general technique for solving first order linear differential equations. \u00a0Let<\/span><\/i>\u00a0\\(y_{sep}(t)\\)<\/span>\u00a0be the solution for an initial value problem using separation of variables, and let \\(y_{lin}(t)\\)\u00a0<\/span><\/i>be the solution for the same initial value problem using the technique for linear differential equations. Which of the following statements correctly states the relationship between<\/span><\/i>\u00a0\\(y_{sep}(t)\\)<\/span>\u00a0and \\(y_{lin}(t)\\)<\/span><\/i>?<\/span><\/i><\/p>\n