A Framework for Integrating Conceptual and Procedural Understanding in the First Two Years of Undergraduate Mathematics

By Karen Keene and Nicholas Fortune, North Carolina State University

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.

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).

However, some researchers, including ourselves, provide evidence that “teaching the procedures to solve problems and find solutions” and “providing ways for teaching concepts first so students will truly understand” 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 Framework for Relational Understanding of Procedures. It was developed as part of Rasmussen and colleagues’ work in differential equations teaching and learning (Rasmussen et. al., 2006). Skemp coined the original definition; she defines relational understanding as “knowing both what to do and why” and contrasts it to instrumental understanding as “rules without reason” (1976, p. 21).

Following, we describe the six components of the Framework for Relational Understanding of Procedures. The idea is that each category can be used to consider and enhance students’ learning as they study a procedure.  For 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.  Likely, each instructor could add other algorithms in differential equations as well as other courses.

Components of Relational Understanding of Procedures

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.  

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?

Example:

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.  Let \(y_{sep}(t)\) be the solution for an initial value problem using separation of variables, and let \(y_{lin}(t)\) 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 \(y_{sep}(t)\) and \(y_{lin}(t)\)?

  • \(y_{sep}(t)\) is not equivalent to \(y_{lin}(t)\)
  • \(y_{sep}(t)\) is equivalent to \(y_{lin}(t)\) for all t
  • \(y_{sep}(t) = y_{lin}(t)\) only for equilibrium solutions
  • \(y_{sep}(t) = y_{lin}(t)\) only at the initial condition

Student can identify when it is appropriate to use a specific procedure.

Students often can do the procedure when they know that is what is needed. However, they often are unable to decide before they start which procedure is needed. Ultimately, one reason that this is an issue is because of the structure of typical textbooks (e.g., the homework always matches the section). How many of you have had students say, “I could do all the problems in the homework, but then I didn’t know what to do for the exam”?  

Example:

Circle all that apply.  A differential equation can be solved with the technique for first order linear ODEs if:

  • it has the form \(\frac{dy}{dx}=ax+by\) for some constants a and b
  • it has a solution whose graph is linear
  • it has the form  \(\frac{dy}{dx}=f(x)y+g(x)\) for some functions f(x) and g(x)
  • it has the form \(\frac{dy}{dx}=mx+b^2\)

Student can correctly carry out the entire procedure or a selected step in the procedure.

This is what we typically think of as doing a problem, or performing the framework. Can the student do the steps necessary to complete a problem correctly? Can the student analyze where they are in the procedure and know what to do next?

Example:

A student is solving a first order linear differential equation and at some point in her solution process she correctly gets the expression to \(e^{2y}\left(\frac{dy}{dt}+2y\right)\). This expression is equivalent to which of the following?

a) \((e^{2t}y)’\)

b) \((e^{3t}y)’\)

c) \(e^{3t}y’\)

d) \(e^{2t}y’\)

Student understands the reasons why a procedure works overall. Additionally, student knows the motivation or rationale for key steps in the procedure.

This step fundamentally involves the conceptual idea behind the procedure. As instructors, we make efforts to teach these ideas in our classes on a regular basis.  However, are we concerned about how the students grow to understand the “why” of the procedure? Do the reasons for the steps play a part in the students’ solving? Can the students go back and make modifications because they understand what is really happening?

Example:

Which of the following would be a justification for one or more of the steps needed to solve a first order linear differential equations? Circle all that apply.

  • Fundamental Theorem of Calculus
  • Mean Value Theorem
  • L’Hopital’s Rule
  • Product Rule

Student can symbolically or graphically verify the correctness or reasonableness of a purported outcome to a procedure without repeating the procedure.

This component is about thinking through the answer in a way that you can decide if it makes sense. Our experience says that if you ask students to check for the reasonableness, they often just repeat the procedure, and this indicates a need to push for the bigger picture of making sense of a solution beyond just doing. Showing competence in this component might involve either checking in terms of seeing if the solution works, or using a graphical or numerical technique to see if the two solutions are compatible. Can the student find a way to check for correctness? Can the student decide if answers are reasonable?

Example:

Joey is solving an autonomous differential equation of the form \(\frac{dy}{dt}=f(x)\), using separation of variables to find the general solution. At one point in his solution process he correctly gets \(e^x=t^2+c \) .  His final answer is then \(x=ln(t^2)+c \). We can verify that Joey’s final answer is:

a)   Correct because \(x=ln(t^2)+c \) says that graphs of solutions are shifts of each other along the t axis (that is, they are horizontal shifts of each other).

b)   Correct because \(x=ln(t^2)+c \) says that graphs of solution are shifts of each other along the x axis (that is, they are vertical shifts of each other).

c) Incorrect because \(x=ln(t^2)+c \) says that graphs of solutions are shifts of each other along the t axis (that is, they are horizontal shifts of each other).

d) Incorrect because \(x=ln(t^2)+c \) says that graphs of solutions are shifts of each other along the x axis (that is, they are vertical shifts of each other).

e)   Incorrect because \(e^x\) is always positive.

Student can make connections within and across representations involved in the problem and solution: symbolic, graphical, and numerical.

Educational literature suggests that one way to demonstrate deep understanding is to make connections among representations. Traditionally, in upper level mathematics, the representations are often symbolic, but in differential equations, linear algebra, and other freshman and sophomore classes, there are several representations, and students who can be flexible and move among them have better understanding.

Example:

Jung Hee uses a slope field to determine the long term behavior (that is what happens as \(t \to \infty \)) of the solution to the initial value problem \(\frac{dy}{dt}=0.4y(70-y) \).  Which of the following methods could be used to corroborate the long term behavior she found by using the slope field? Circle all that apply.

  • The technique to solve separable differential equations.
  • Euler’s numerical method with a small step size.
  • The technique to solve first order linear differential equations.
  •  None of the above.

Conclusion

The framework described here and the examples from an assessment developed for relational understanding (Keene, Glass, Kim, 2011) may offer some ways to think about teaching procedures that are the foundation of many of the early undergraduate mathematics class. It may not be a matter of trying to teach the procedures or the concepts (as a dichotomy) but of developing a relational understanding of the procedures so that students can not only find answers, but also understand the underpinnings and development of the procedures.  We believe that if students have this relational understanding, not only will they perform better in their classes, they will retain the skills and understandings over periods of time.  This will result in students doing better in all their mathematics classes.

We would like to acknowledge Dr. Chris Rasmussen for his contributions to the work.

References

Haapasalo, L., & Kadijevich, D. (2000). Two types of mathematical knowledge and their relation. Journal für Mathematik-Didaktik, 21(2), 139-157.

Hassenbrank, J. & Hodgson., T. (2007).  A framework for developing algebraic understanding & procedural skill: An initial assessment. In Proceedings of Research in Undergraduate Mathematics Annual Conference.

Hiebert, J. (2013). Conceptual and procedural knowledge: The case of mathematics. Routledge.

Keene, K. A., Glass, M. & Kim, J. H. (2011). Identifying and assessing relational understanding in ordinary differential equations. In Proceedings of the 41st Annual Frontiers in Education Conference, Rapid City, SD.

Rasmussen, C., Kwon, O., Allen, K., Marrongelle, K. & Burtch, M. (2006). Capitalizing on advances in K-12 mathematics education in undergraduate mathematics: An inquiry-oriented approach to differential equations. Asia Pacific Education Review, 7, 85-93.

Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26.
Star, J. R. (2005). Reconceptualizing procedural knowledge.  Journal for Research in Mathematics Education, 36(5), 404-415.

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