Procedures vs Understanding in Math

This article is excerpted from “Math Education in the U.S.” (Chapter 10)

I went to school in the 50’s and 60’s when students first learned how to add and subtract in second grade. After spending some time memorizing the basic addition and subtraction facts and learning how to add and subtract single digit numbers, I was excited to hear my teacher announce one day that we would now learn how to solve problems like 43 + 52 and 95 – 64. In teaching the method, the teacher explained how the procedure relied on place value–what the ones place and tens place meant.  I became bored with the explanation, began to daydream and missed the description of the procedure. The teacher then announced that we would now take a test on what we had just learned.

Faced with having to solve ten two-digit addition problems, I fell quickly behind the rest of the class. The teacher announced that she would not go on until everyone turned in their test. Students now put pressure on me as I desperately drew sticks on the side of my paper to “count up” to the answers. Finally, a girl across from me whispered “Add the ones column first and then the tens.” This advice made perfect sense to me and I finished the problems quickly. Although I had missed the explanation of why the ones and tens columns were added separately, it wasn’t long until I understood why after hearing the explanation again when the time came for learning how to “carry”. I was now receptive to what was going on with the procedure.

Procedures as “Magic Corridors” to Understanding

The issue of balance between procedural fluency and conceptual understanding continues to dominate discussions within the education community. The vignette above illustrates how procedural fluency may lead to understanding. This is true for all students, but is particularly relevant for students who may have learning disabilities.

Such students may find contextual explanations burdensome and hard to follow, resulting in feelings of frustration and inadequacy. It is not unusual in the lower grades for LD students–as well as non-LD students–to become impatient and wish that teachers would “just tell me how to do it.”

For many students, the “why” of the procedure is easier to navigate once fluency is developed for the particular procedure. The reason for this is given in large part through Cognitive Load Theory (Sweller, et al, 1994), which states that working memory gets overloaded quickly when trying to juggle many things at once before achieving automaticity of certain procedures.

An example of this is the plight of a visitor to a new city trying to find his way around. In getting from Point A to Point B, the visitor may be given instruction that consists of taking main roads; the route is simple enough so that he is not overburdened by complex instructions. In fact, well-meaning advice on shortcuts and alternative back roads may cause confusion and is often resisted by the visitor, who when unsure of himself insists on the “tried and true” method.

The visitor views these main routes as magic corridors that get him from Point A to B easily. He may have no idea how they connect with other streets, what direction they’re going, or other attributes. With time, after using these magic corridors, the visitor begins see the big picture and notices how various streets intersect with the road he has been taking. He may now even be aware of how the roads curve and change direction, when at first he thought of them as more or less straight. The increased comfort and familiarity the visitor now has brings with it an increased receptivity to learning about–and trying–alternative routes and shortcuts. In some instances he may even have gained enough confidence to discover some paths on his own.

In math, learning a procedure or skill is a combination of big picture understanding and procedural details. Research by Rittle-Johnson et. al., (2001) supports a strong interaction between understanding and procedures and that the push-pull relationship is necessary. Daniel Ansari (2011), a leading scholar of cognitive developmental psychology who studies brain activity during the learning of mathematics, also maintains that neither skill nor understanding should be underemphasized—they provide mutual scaffolding and both are essential.

Sometimes understanding comes before learning the procedure, sometimes afterward. The important point is recognizing when students are going to be receptive to learning the big picture understandings about what is really happening when they perform a procedure or solve a particular type of problem. Like visitors to a strange city, for many students, understanding comes after some degree of mastery of a particular skill or procedure.

For students with learning disabilities, providing explicit instruction on the procedure should take precedence. A study by Morgan (2014) indicates that direct and explicit instruction given to first grade students with learning disabilities in math has positive effects. Conversely, the study shows that student-centered activities (such as manipulatives, calculators, movement and music) did not result in achievement gains by such students. Of particular significance is that the study also found that direct and explicit instruction benefited those students without learning disabilities in math. To this end, we would add that an undue emphasis on conceptual understanding can decrease the amount of needed explicit instruction for students.

For many concepts in elementary math, it is the skill or procedure itself upon which understanding is built. The child develops his or her understanding by repeatedly practicing the pure skill until it is realized conceptually through familiarity and tactile experience that forges pathways and connections in the brain.  But in terms of sequential priority, there is no chicken-and-egg problem: more often than not, skill must come first, because it is difficult to develop understanding in a vacuum. Procedural fluency provides the appropriate context within which understanding can be developed. It is important to note, however, that for everyone, there may be certain procedures that will never be fully understood. And that’s OK. This idea that one hundred percent of all students must understand one hundred percent of all the concepts behind one hundred percent of all procedures is one hundred percent unrealistic, misguided and damaging. What is key is the level of understanding achieved, and whether a student can recognize what is needed to solve a problem, and execute the procedures necessary to solve it.

This is not to say that the conceptual underpinning should be omitted when teaching a procedure or skill. But while some explanation of the context is necessary to motivate the procedure, the issue is the how much. Students with learning disabilities should be given explanations of how to proceed sooner rather than later. As discussed in more detail in the next section, after the standard procedure(s) are mastered alternative methods designed to provide deeper understanding of the concepts behind the procedure can then be provided when students are more receptive to such alternatives. Providing the alternatives too soon can result in the student becoming overloaded with information (called cognitive overload). It is also important to recognize that there will be some students who may not fully comprehend the concepts behind a procedure or problem solving technique at the same pace as others in their cohort.

Robert Craigen, a professor of mathematics at University of Manitoba describes the thinking behind delaying the teaching of standard procedures and algorithms: “I showed some student work in which an easy two-line calculation was performed by way of convoluted and disorganized ramblings, tallies and obtuse sketches to an education ministry consultant. His response was ‘See! They are showing understanding!’ ”

Worked Examples and Scaffolding

In teaching procedures for solving both word problems and numeric-only problems, an effective practice is one in which students imitate the techniques illustrated in a worked example. (Sweller, 2006). Subsequent problems given in class or in homework assignments progress to variants of the original problem that require them to stretch beyond the temporary support provided by the initial worked example; i.e., by “scaffolding”.

Scaffolding is a process in which students are given problems that become increasingly more challenging, and for which temporary supports are removed. In so doing, students gain proficiency at one level of problem-solving which serves to both build confidence and prepare them for a subsequent leap in difficulty.  For example, an initial worked example may be “John has 13 marbles and gives away 8. How many does he have left?” The process is simple subtraction. A variant of the original problem may be: “John has 13 marbles.  He lost 3 but a friend gave him 4 new ones.  How many marbles does he now have?”  Subsequent variants may include problems like “John has 14 marbles and Tom has 5. After John gives 3 of his marbles to Tom, how many do each of them now have?”

Procedure versus “Rote Understanding”

It has long been held that for students with learning disabilities, explicit, teacher-directed instruction is the most effective method of teaching. The final report of the President’s National Math Advisory Panel (2008) states: “Explicit instruction with students who have mathematical difficulties has shown consistently positive effects on performance with word problems and computation. Results are consistent for students with learning disabilities, as well as other students who perform in the lowest third of a typical class.” (p. xxiii). These statements have been recently confirmed by Morgan, et. al. (2014) cited earlier. The treatment for low achieving, learning disabled and otherwise struggling students in math thus includes memorization and other explicit instructional methods.

Currently, with the adoption and implementation of the Common Core math standards, the prevailing interpretation of the standards has resulted in an increased emphasis on students understanding the conceptual underpinnings of algorithms and problem-solving procedures. Instead of adding multi-digit numbers using the standard algorithm and learning alternative strategies after mastery of that algorithm is achieved (as we earlier recommended be done), students must do the opposite. That is, they are required to use inefficient strategies that purport to provide the “deep understanding” when they are finally taught to use the more efficient standard algorithm. The prevailing belief is that to do otherwise is to teach by rote, and with no understanding. Students are also being taught to reproduce explanations that make it appear they possess understanding—and more importantly, to make such demonstrations on the standardized tests that require them to do so.  

Such an approach is tantamount to saying, “If we can just get them to do things that look like what we imagine a mathematician does, then they will be real mathematicians.” Forcing students to think of multiple ways to solve a problem, for example, or to write an explanation for how they solved a problem or why something works does not in and of itself cause understanding. It is investment in the wrong thing at the wrong time.

The “explanations” most often will have little mathematical value and are naïve because students don’t know the subject matter well enough. The result is at best a demonstration of “rote understanding”–a student engaging in the exercise of guessing (or learning) what the teacher wants to hear and repeating it. At worst, it undermines the procedural fluency that students need.

Forcing students to think of multiple ways to solve a problem, for example, or to write an explanation for how they solved a problem or why something works does not in and of itself cause understanding. It is investment in the wrong thing at the wrong time.

Understanding, critical thinking, and problem solving come when students can draw on a strong foundation of domain content relevant to the topic being learned. As students (non-LD as well as LD) establish a larger repertoire of mastered knowledge and methods, the more articulate they become in explanations.

While some educators argue that procedures and standard algorithms are “rote”, they fail to see that exercising procedures to solve problems requires reasoning with such procedures – which in itself is a form of understanding.  This form of understanding is particularly significant for students with LD, and definitely more useful than requiring explanations that students do not understand for procedures they cannot perform.

References

Ansari, D. (2011). Disorders of the mathematical brain : Developmental dyscalculia and mathematics anxiety. Presented at The Art and Science of Math Education, University of Winnipeg, November 19th 2011. http://mathstats.uwinnipeg.ca/mathedconference/talks/Daniel-Ansari.pdf

Morgan, P., Farkas, G., MacZuga, S. (2014). Which instructional practices most help first-grade students with and without mathematics difficulties? Educational Evaluation and Policy Analysis Monthly 201X, Vol. XX, No. X, pp. 1–22. doi: 10.3102/0162373714536608

National Mathematics Advisory Panel. (2008). Foundations of success: Final report. U.S. Department of Education.

Rittle-Johnson, B., Siegler, R.S., Alibali, M.W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, Vol. 93, No. 2, 346-362. doi: 10.1037//0022-0063.93.2.346

Sweller, P. (1994) Cognitive load theory, learning difficulty, and instructional design. Learning and Instruction, Vol. 4, pp. 293-312

Sweller, P. (2006). The worked example effect and human cognition. Learning and Instruction, 16(2) 165–169

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