Life is full of alternatives. Would like fries or coleslaw with your meal? Should you put on your right or your left shoe first? Should you attempt to solve a math problem using algebraic procedures, or your intuitive sense of the situation?
Life is also full of false choices: there are many occasions when you do not have to make a choice unless you wish to. You could have fries with a side order of coleslaw. If you wear loafers, you could slide your feet into both shoes at the same time. And many math problems can be solved quite successfully using a combination of intuitive reasoning and algebraic procedures.
Many high school students (and beyond) seem to perceive Procedural vs Intuitive approaches to be mutually exclusive. This is learned behavior, because elementary school students are often taught a mixture of intuitive (number sense) and procedural (adding a column of numbers) topics, and middle school students learn the beginnings of algebraic procedures while also often encouraged to pursue intuitive solutions to a variety of problems. This seems to be done because a number of students who struggle with procedural approaches to math problems may be successful with intuitive ones.
But high School students are often strongly encouraged (either directly or indirectly) to use only procedural approaches, probably because there is a high probability that intuitive approaches will become increasingly difficult to apply as the assigned problems become more complex. Many students who have relied on intuitive approaches quite successfully for nine years find themselves suddenly being asked (or forced) to abandon trusted approaches in favor of pushing a bunch of symbols around an equal sign. Needless to say, they often do not relish the prospect.
Concepts are both the foundation for, and the link between, intuitive and procedural approaches. They help explain why each procedure or intuitive approach works. Some students can grasp the conceptual side of algebra at a young age, while for others it can be a struggle (even if they are good intuitive problem solvers).
When I meet with a high school math student for the first time, I have found that most focus primarily on procedure. Seldom do they raise conceptual questions. It seems this may be the result of high school teachers using a predominantly procedural approach to solving problems, probably because they feel they can get students to master it more efficiently. However, will students still have it mastered when the quiz or test occurs?
High school students perceive that intuitive approaches to problems are being discouraged, and that procedural approaches are being mandated. Some teachers emphasize the concepts behind the procedures more than others, leading students to wonder if it is important material or not. And lastly, few assessments address conceptual understanding during the course of each year. “Procedures rule… so just suck it up and memorize them” – that seems to be the way many students perceive things.
As a math tutor, I have met many students who seem to have reached their “memorization limit” in math. Their minds are filled with a profusion of formulae, and they increasingly can neither remember nor figure out which one to use for the problem at hand. While furiously memorizing for the test may have produced good grades for much of their school career to date, this approach begins to fray at the edges over time. Eventually, a crisis point is reached. Grades are not what they once were. Confidence sags. The subject is no longer fun. And college application time is looming large.
My first tack with such students is to try to entice them away from purely procedural approaches. If I can get them to talk about and doodle the ideas and concepts behind each problem, they often find conceptual connections that they had not perceived earlier and solve the problem on their own. This helps them to begin to regain their confidence. As their conceptual and intuitive skills are re-engaged, the pile of memorized procedures begins to feel more ordered and manageable – in part because they begin to see the relationships between many procedures: one is a subset of another, etc.
Reviewing the whole chapter briefly in one sitting, instead of just the most recent section covered in class, often helps engage the intuition if I try to emphasize the connections between what was introduced in each section. Procedure and concept become better linked. It is also very helpful to point out connections between a topic and those that have preceded or will follow.
There is great value to briefly (10 to 15 minutes) summarizing “linear equations”, or “quadratics”, or “inverse functions” from a conceptual (hand-waving and doodling, not formula solving) perspective. However, this value is often more apparent after having introduced much of the material procedurally. In other words, I encourage math teachers to “step back” at the end of each chapter and provide an overview and reiteration of the entire chapter (or perhaps even several chapters) from a more conceptual perspective.
We should teach both intuitive and procedural approaches to all students whenever possible. The two approaches re-enforce one another. Good problem solvers need an arsenal of problem solving tools that they trust, some procedural, some intuitive. I believe that good problems solvers also rely on both intuitive and procedural approaches in parallel while solving many problems – each serves to double check the results of the other, and helps students avoid “the usual” errors (sign errors, distribution errors, etc.) while working a problem.
We should also devote more time (the most scarce of commodities in a classroom) to helping all students master the concepts behind what they are learning by asking questions that require conceptual answers. While different students may require greatly varying amounts of time to master a concept, doing so will have a profound impact on their math grades throughout High School. Formulae that were vaguely remembered and not really understood can become intuitively “obvious”. And when that happens, self-confidence soars, communication skills improve, and (cue the choir and rainbow) mathematics can once again be fun for both teacher and student.
Procedural vs Intuitive need not be an either-or choice. Both approaches are important, and they would hopefully command equal respect in every math classroom. Furthermore, we should expect students to master both to the best of their abilities, since both are often required when working on challenging problems in both math class and life.
However, students need not master both simultaneously… most (but definitely not all) students probably are used to a procedural introduction, time to build a base of experience, then hopefully a conceptual re-introduction that connects these processes to others they may already know, and perhaps a final intuitive brain-storming session which gives them the opportunity to connect process, concept, and intuition. Then again, some students may prefer the reverse sequence.
For teachers looking to ask open-ended questions that can help engage their students’ intuitive thinking, 26 Questions You Can Ask Instead has some fantastic suggestions that can help spark student responses and conversation instead of silence.
For a discussion of some of the mechanisms at work in our understanding of mathematics, I highly recommend Richard Skemp’s 1976 article: Relational Understanding and Instrumental Understanding.