## INTRODUCTION

The science education literature on quantitative problem solving emphasizes the importance of incorporating conceptual reasoning in two phases of problem solving: (1) initial qualitative analysis of the problem situation to determine the relevant mathematical equations and (2) interpretation of the final mathematical answer, to check for physical meaning and plausibility (Heller, Keith, & Anderson, 1992; Redish & Smith, 2008; Reif, 2008). Without disputing the importance of these phases of problem solving, we note that almost no research has focused on the “mathematical processing” stage where the equations are used to obtain a solution. In this paper, we investigate different ways that students can process equations while problem solving. We argue that a feature of problem-solving expertise—and a feasible instructional target in physics, chemistry, and engineering courses—is *blended processing*, the opportunistic *blending* of formal mathematical and conceptual reasoning (Fauconnier & Turner, 2003; Sherin, 2001) *during* the mathematical processing stage. In other words, we argue that expert problem solving involves exploiting opportunities to use conceptual reasoning to facilitate the manipulation of equations themselves.

To make our case, we first review how physics education researchers have conceptualized and taught quantitative problem solving. We then discuss research suggesting the importance of blending conceptual reasoning with symbolic manipulations in quantitative problem solving, and we propose *symbolic forms* (Sherin, 2001) as cognitive resources that facilitate such *blended processing*. Then we use contrasting case studies of two students solving a physics problem, to illustrate what we mean by blended processing. Alex solves the problem by representing the physical situation with a diagram, identifying the relevant physics equations, using those equations to compute a numerical answer and reflecting upon that answer—in accord with problem-solving procedures taught in physics classrooms (e.g., Giancoli, 2008; Young & Freedman, 2003) and advocated in education research (Heller et al., 1992; Huffman, 1997; Reif, 2008; Van Heuvelen, 1991a). Pat, by contrast, blends symbolic equations with conceptual reasoning about physical processes to find a “shortcut” solution. After analyzing Alex's and Pat's responses in detail, we show that some other introductory physics students in our data corpus also do the type of blended processing done by Pat. In documenting what such blended processing can look like for undergraduate students in an introductory physics course, we make the case that (1) the opportunistic use of blended processing is part of quantitative problem-solving expertise, (2) a theoretical construct called *symbolic forms* (Sherin, 2001) contributes to a good cognitive account of Pat's (and the Pat-like students’) blended processing, and (3) such blended processing is a feasible instructional target in science and engineering courses.

### LITERATURE REVIEW: CONCEPTUALIZATIONS OF EXPERT PROBLEM SOLVING

In this section, we present a common conceptualization of expertise in quantitative physics problem solving, as well as challenges to a particular aspect of that conceptualization. We limit our discussion to *quantitative* problem solving, because our argument specifically concerns the processing of equations in problem solving.

### Research on Expert Problem Solving and Resulting Instructional Strategies Emphasize an Initial Conceptual Reasoning Phase

As a central feature of their professional practice, scientists apply domain-specific knowledge to solve quantitative problems (Redish & Smith, 2008; Reif, 2008; Reif & Heller, 1982). Partly for this reason, developing problem-solving expertise in students has become a central concern of science education researchers and practitioners (Hsu, Brewe, Foster, & Harper, 2004; Maloney, 1994, 2011).

Early research on physics problem solving suggests a difference between experts and novices. Experts tend to start with a conceptual analysis of the physical scenario, which then leads into the mathematics. By contrast, novices tend to start by selecting and manipulating equations that include relevant known and unknown quantities (Larkin, McDermott, Simon, & Simon, 1980; Simon & Simon, 1978). Specifically, on standard textbook physics problems, experts cue into relevant physics principles whereas novices cue into surface features and their related equations (Chi, Feltovich, & Glaser, 1981). Building on these findings, subsequent research has explored the benefits of helping students analyze the problem situation conceptually (Dufresne, Gerace, Hardiman, & Mestre, 1992; Larkin & Reif, 1979) and has incorporated initial conceptual thinking into models of effective quantitative problem solving (Heller & Reif, 1984; Reif & Heller, 1982).

This research on expert–novice differences has also influenced researchers’ formulation of multistep problem-solving procedures intended for students to learn and apply (Heller et al., 1992; Huffman, 1997; Reif, 2008; Van Heuvelen, 1991a, 1991b). These procedures generally include versions of the following steps: (1) perform an initial conceptual analysis using relevant physics principles, (2) use this qualitative analysis to generate the relevant mathematical equations, (3) use equations to obtain a mathematical solution in a “mathematical processing” step, and (4) interpret that mathematical solution in terms of the physical scenario. These procedures incorporate the expert–novice findings by encouraging students to reason conceptually before jumping into mathematical manipulations.

In these strategies, the steps are meant to mirror behaviors exhibited by experts while also remaining accessible enough to be instructional targets. The explicit teaching and enforcement (through grading policies) of these problem-solving procedures has increased the quality and frequency of physical representations used in problem solving, as well as the correctness of students’ answers, in comparison with traditional instruction (Heller et al., 1992; Huffman, 1997; Van Heuvelen, 1991a).

### Studies of Quantitative Problem Solving Have Not Focused on *How* Equations Are Processed to Reach Solutions

The studies described above illustrate a common feature of research on students’ quantitative problem solving: to the extent these studies focus on equations, they focus on how students *select* equations rather than on how students *use* those equations after their selection. While this focus has produced important findings and implications for instruction (e.g., emphasizing initial conceptual reasoning for selecting relevant equations), it has also limited attention to how students process mathematical equations to obtain numerical or symbolic solutions.

In some research, the equations are treated (either explicitly or implicitly) as computational tools, devices to find unknown values from known values through symbolic and numeric manipulation. This is true of the problem-solving procedures described above (Heller et al., 1992; Huffman, 1997; Van Heuvelen, 1991a) and of studies on how successful problem solvers use mathematics (e.g., Dhillon, 1998; Taasoobshirazi & Glynn, 2009).

Other more recent studies have not attended to any aspect of how equations are processed. Walsh, Howard, and Bowe (2007) focused mainly on how students *selected* relevant equations rather than on how those equations are subsequently used. Some studies have deemphasized the use of mathematics completely and focused only on students’ qualitative analysis, both in instructional interventions (e.g., Mualem & Eylon, 2010) and in finding predictors of problem-solving expertise (e.g., Shin, Jonassen, & McGee, 2003).

The first author (Kuo) did a search through *Science Education*, *Journal of Research in Science Teaching*, *Research in Science Education*, *International Journal of Science Education*, *American Journal of Physics*, *The Journal of Engineering Education*, *Cognition & Instruction*, and *Journal of the Learning Sciences* from January 2000 to March 2012 and *Physical Review Special Topics—Physics Education Research* from July 2005 (its inception) to March 2012. He looked for articles focusing explicitly on problem solving in which the analysis attended to the possibility of processing equations in multiple ways. First, the titles of all articles were scanned and abstracts of articles with titles containing terms such as “problem solving” or “equations” were read. If the abstract described investigations of components of problem-solving expertise, the article itself was read. This search found no studies that focused upon the mathematical processing step in quantitative problem solving or described alternatives to using equations as computational tools.

### Other Studies Suggest the Importance of Blending Conceptual Reasoning With Symbolic Manipulations

We have shown that research on quantitative problem solving has not attended to different ways that mathematical equations can be used to obtain numerical or symbolic solutions. This paper presents two different ways that such equations may be used: (i) as computational tools, manipulated to solve for unknown quantities; or (ii) blended with conceptual meaning to produce solutions (or progress toward solutions). But why is this difference significant?

Other pockets of research suggest that using equations without looking to their conceptual meaning *during the processing* can, in certain situations, reflect a *lack* of expertise. In mathematics education research, for example, Wertheimer (1959) asked students to solve problems of the following type: (815 + 815 + 815 + 815 + 815)/5 = ?. Students who solved the problem by computing the sum in the numerator and then dividing by 5 missed a possible shortcut around explicit computation: using the underlying conceptual meanings of addition and division to realize that the solution is 815, without doing any computations. Students who missed the shortcut had demonstrated proficiency with the mathematical procedures, but not understanding of the underlying conceptual meaning. In addition, Arcavi (1994) suggested the importance of *symbol sense*: an ability to reason conceptually about symbols. This *symbol sense* includes the ability to interpret the conceptual meaning behind symbolic relationships, generate expressions from intuitive and conceptual understanding, and decide when and how best to exploit one's conceptual understanding of symbols.

Redish and Smith (2008), writing about expert problem solving in science and engineering, also challenged the view that symbolic manipulation should be *a priori* divorced from conceptual reasoning, saying “… because of the fact that the equations are physical rather than purely mathematical, the processing can be affected by physical interpretations” (Redish & Smith, 2008, p. 302). Just as Wertheimer showed that students’ conceptual understanding of *mathematical* operations influenced how they carry out calculations in an arithmetic problem, Redish and Smith suggest that students’ interpretations of equations in terms of the physical scenario can influence how they use the equations in solving physics/engineering problems.

Again, we do not dispute the instructional value of prior research on problem-solving procedures that emphasize conceptual reasoning at the start and the end of problem solving. As noted above, an instructional emphasis on such procedures has helped students to produce more and better representations and to produce correct solutions more frequently. However, Wertheimer, Arcavi, and Redish and Smith suggest the importance of focusing on *how* students process equations in their quantitative problem solving, a focus not present in the quantitative problem-solving literature. Specifically, these researchers argue that blending conceptual reasoning with mathematical formalism in the processing of equations—what we refer to as *blended processing*—may be productive and reflect greater expertise in some situations than using an equation simply as a computational tool. In the next section, we discuss the use of *symbolic forms,* which we argue is one specific way that blended processing can occur in physics problem solving.

### Symbolic Forms: A Blend of Conceptual Reasoning and Mathematical Formalism

In arguing that problem solving does not necessarily proceed from direct application of canonical physics principles, Sherin (2001) proposed the existence of knowledge structures called *symbolic forms,* which link mathematical equations to intuitive conceptual ideas. Specifically, in a *symbolic form,* a *symbol template* is blended with a *conceptual schema*.

A *symbol template* represents the general structure of a mathematical expression without specifying the values or variables. For example, □ = □ is the symbol template for Newton's second law (*F = ma*), whereas the symbol template for the first law of thermodynamics, ∆*E* = *Q* + *W*, is □ = □ + □. Each symbol template is not unique to a single equation. For instance, the symbol template □ + □ + □ can describe both the expression *x*_{0} + *v*_{0}*t* + 1/2*at*^{2} and the expression *P*_{0} + 1/2*ρv*^{2} + ρ*gh*.

A *conceptual schema* is an intuitive idea or meaning that *can* be (but does not *have* to be) represented in a mathematical equation or expression. By “intuitive” ideas, we mean ideas that are informal and drawn from everyday (nonacademic) knowledge—ideas that make quick and immediate sense and that do not seem to require further explanation. One example of such a conceptual schema is the idea that *a whole consists of many parts*. For example, an automobile can be seen as an assembled whole of many parts such as the engine, the transmission, and the chassis; a wedding guest list can be conceptualized as consisting of the close relatives, the close friends, business contacts, and others; an essay might be viewed as the compilation of the introduction, the main body of argument, and the conclusion. Similarly, a physics student's conceptual understanding of the total mechanical energy of a system may be grounded in the idea that it consists of many different types of energy: kinetic, gravitational potential, spring potential, and so on.

Another conceptual schema, applicable to reasoning about a game of tug of war or about a marriage between a spendthrift and a miser, is the idea of *opposing influences*. In physics, this conceptual schema may apply to a student's conceptual understanding of a falling object, where air resistance opposes the influence of gravity (Sherin, 2001).

As these examples illustrate, a conceptual schema in Sherin's framework is an *intuitive* idea used in everyday, nonscientific reasoning, not a formal scientific concept. A student's understanding of a formal scientific concept (such as mechanical energy) can draw upon these intuitive conceptual schemata (such as *a whole consists of many parts*); but the conceptual schema also plays a role in students’ reasoning about other subjects, such as wedding guest lists.

A *symbolic form* is a cognitive element that blends a symbol template with a conceptual schema, such that the equation is interpreted as expressing meaning corresponding to the conceptual schema. For example, the *Parts-of-a-Whole* symbolic form blends the symbol template “□ + □ + □ +···” with the conceptual schema of *a whole consisting of many parts*. The boxes take on the meaning of “parts” that add up to represent a “whole.” A student who uses the *Parts-of-a-Whole symbolic form* to interpret the equation *E* = 1/2*mv*^{2} + mgh +1/2*kx*^{2} would say that the overall (whole) energy of the system consists of the sum of three separate parts (kinetic, gravitational potential, and spring potential). So, when a symbolic form is used, the reasoning is neither purely formal mathematical nor purely conceptual; it is blended into a unified way of thinking that leverages *both* intuitive conceptual reasoning and mathematical formalism. By contrast, a writer thinking about her essay might think of the parts of her essay (Introduction, etc.) using the conceptual schema corresponding to *Parts-of-a-Whole*, but is unlikely to think of adding those parts in an equation. Only when a conceptual schema is blended with an equation's symbol template is a symbolic form present in the person's reasoning.

Sherin (2001) observed students productively using symbolic forms in two ways. One was to produce novel equations from an intuitive conceptual understanding of a physical situation. For example, figuring out how much rain would hit them in a rainstorm, a pair of students wrote the equation [total rain] = [# raindrops/s] + *C*. Their explanation of this equation reflected the use of *Parts-of-a-Whole*. Specifically, they said the total rain would come from two sources: the amount falling on top of the person, indicated by [# raindrops/s], and the amount striking the front of the person as they walk forward, indicated by *C*. Other research has also supported the explanatory power of symbolic forms in models of how students translate physical understandings into mathematical equations (Hestenes, 2010; Izsák, 2004; Tuminaro & Redish, 2007).

The other way in which Sherin's subjects used symbolic forms was to interpret mathematical equations in terms of a physical scenario, using functional relations expressed by the equation. For example, after deriving the terminal velocity of a falling object, *v = mg/k*, several students noticed that the mass, *m*, was in the numerator. Students interpreted this as meaning that a heavier object reaches a greater terminal velocity. Sherin modeled these students as using the *Prop*+ (positive proportionality) symbolic form—a blend of the symbol template […*x*…/…] with the conceptual schema that one quantity increases as another one increases—to read out a physical dependence from the mathematical equation. Later, Sherin (2006) hypothesized that *Prop*+ was tied to physical notions of effort and agency, what we see as “cause and effect.” Other researchers have also used symbolic forms to model how students translate from mathematical solutions into physical understanding (Hestenes, 2010; Tuminaro & Redish, 2007; VanLehn & van de Sande, 2009).

These two ways in which Sherin (2001) saw students use symbolic forms correspond roughly to the two “steps” involving conceptual reasoning, as described by the problem-solving literature: (i) translating conceptual understanding of a physical scenario into mathematical equation(s) at the start of problem solving and (ii) giving a physical interpretation of a mathematical solution at the end.1 We have seen no studies that look at how symbolic forms–based reasoning—or blended conceptual and formal mathematical reasoning more generally—might enter into the “mathematical processing” step in quantitative problem solving.

### Building on the Literature to Explore the Mathematical Processing Step

We contribute to the literature on quantitative problem solving in three ways. First, we focus on different ways in which mathematical equations can be used to reach a solution. This is a relatively unexplored topic, as discussed above. We find that equations can be used as tools for symbolic or numerical manipulations or as tools in blended processing. As we argue, the opportunistic use of such blended processing can reflect greater expertise than symbolic or numerical computation (Arcavi, 1994; Redish & Smith, 2008; Wertheimer, 1959).

Second, we argue that symbolic forms help to explain the patterns we document below in students’ problem solving and are therefore productive analytical tools for researchers trying to understand the nature of blended processing.

Third, we argue that symbolic forms are also plausible targets for instruction in introductory physics, partly because they rely on intuitive rather than formal (or discipline-specific) reasoning. Students can therefore productively use symbolic forms *while* they are still learning difficult physics concepts.