Finding the Balance between Procedural Fluency and Conceptual Understanding in Teaching Early Grade Mathematics to Students with Learning Disabilities
By Barry Garelick
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, explicit instruction on procedure should take precedence. A recent study (Morgan, et al, 2014) indicates that direct and explicit instruction given to first grade students with learning disabilities in math has positive effects. Conversely, 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 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 some children, there may be certain procedures for which understanding remains elusive. It is even more important to note that such situation need not prevent such children from performing procedures and solving problems.
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 degree of emphasis. 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. 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 other cohorts.
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?"
Continuing with the example of adding and subtracting, in early grades some students, particularly those with learning disabilities, have difficulty in memorizing the addition and subtraction facts. On top of the memorization difficulties, students then face the additional challenge of applying this knowledge to solving problems. One approach to overcome this difficulty has been used for years in elementary math texts, in which students are provided with a minimum of facts to memorize and then given word problems using only those facts the student has mastered. Such procedure minimizes situations in which working memory encounters interference and becomes overloaded as described in Geary (in press). For example, a student may be tasked with memorizing the fact families for 3 through 5. After initial mastery of these facts, the student is then given word problems that use only those facts. For example, "John has 2 apples and gets 3 more, what is the total?" and "John has some apples and receives 3 more; he now has 5 apples. How many did he have to start with?" Additional fact families can then be added, along with the various types of problems. Applying the new facts (along with the ones mastered previously) then provides a constant reinforcement of memorization of the facts and applications of the problem solving procedures. The word problems themselves should also be scaffolded in increasing difficulty as the student commits more addition and subtraction facts to memory.
Once the foundational skills of addition and subtraction are in place, alternative strategies such as those suggested in Common Core in the earlier grades may now be introduced. One such strategy is known as "making tens" which involves breaking up a sum such as 8 + 6 into smaller sums to "make tens" within it. For example 8 + 6 may be expressed as 8 +2 + 4. To do this, students need to know what numbers may be added to others to make ten. In the above example, they must know that 8 and 2 make ten. The two in this case is obtained by taking (i.e., subtracting) two from the six. Thus 8 + 2 + 4 becomes 10 + 4, creating a short-cut that may be useful to some students. It also reinforces conceptual understandings of how subtraction and addition work .
The strategy itself is not new and has appeared in textbooks for decades. (Figure 1 shows an explanation of this procedure in a third grade arithmetic book by Buswell et. al. (1955).
The difference is that in many schools, Common Core has been interpreted and implemented so that students are being given the strategy prior to learning and mastering the foundational procedures. Insisting on calculations based on the "making tens" and other approaches before mastery of the foundational skills are likely to prove a hindrance, generally for first graders and particularly for students with learning disabilities.
Students who have mastered the basic procedures are now in a better position to try new techniques — and even explore on their own. Teachers should therefore differentiate instruction with care so that those students who are able to use these strategies can do so, but not burden those who have not yet achieved proficiency with the fundamental procedures.
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. A popular textbook on special education (Rosenberg, et. al, 2008) notes that up to 50% of students with learning disabilities have been shown to overcome their learning difficulties when given explicit instruction. 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, there has been increased emphasis and focus on students showing "understanding" of 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 without 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 naÃ¯ve because students don't know the subject matter well enough. The result is at best a demonstration of "rote understanding" — it is 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.
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.
Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He has written a book about his experiences as a long term substitute in a high school and middle school in California: "Teaching Math in the 21st Century".
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
Buswell, G.T., Brownell, W. A., & Sauble, I. (1955). Arithmetic we need; Grade 3. Ginn and Company. New York. p. 68.
Geary, D. C., & Menon, V. (in press). Fact retrieval deficits in mathematical learning disability: Potential contributions of prefrontal-hippocampal functional organization. In M. Vasserman, & W. S. MacAllister (Eds.), The Neuropsychology of Learning Disorders: A Handbook for the Multi-disciplinary Team, New York: Springer
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. https://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
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
Rosenberg, M.S., Westling, D.L., & McLeskey, J. (2008). Special education for today's teachers. Pearson; Merrill, Prentice-Hall. Upper Saddle River, NJ.
Sweller, P. (1994) Cognitive load theory, learning difficulty, and instructional design. Leaming and Instruction, Vol. 4, pp. 293-312
Sweller, P. (2006). The worked example effect and human cognition.Learning and Instruction, 16(2) 165–169