What Did You Get Out of That?
By Dr. Eugene Maier
The topic of discussion at the math education seminar was a research study that compared "outcomes of a traditional and a reform calculus course in terms of students' retention of basic concepts and skills after the passage of time."
The results weren't surprising, seven months after their calculus course was completed, the students in the traditional courses, which stressed procedural skills over concepts, retained more knowledge of skills than concepts. The students in the reformed course, which stressed concepts over procedural skills, retained more knowledge of concepts than skills. However, in seven months time, both groups of students had forgotten most of what they had supposedly learned. This led the researchers to suggest that perhaps we "should spend less time thinking about which type of knowledge to emphasize or which topics to cover and spend more time thinking about how to help students retain more of what they study for a longer period of time."
Several of us thought, extrapolating from our own experiences—both teaching and learning—that without reinforcement or refreshment, most people don't retain over time the specifics of any course they've taken, mathematics or otherwise. So what is it that we can expect people to get out of a math course?
Musing about these matters, I asked my wife at breakfast a couple of days later what she got out of the math course she had taken in college. Her response was immediate: "I got out of taking a lab science." Apparently, it was the lesser of two evils—she either had to take a math class or a lab science to meet the graduation requirements of the School of Journalism and going to a math class 4 times a week was better than going to a science class 3 times a week and spending another 3 hours in a lab.
Struck by her immediate and definite, albeit unexpected, response, I wondered if others would so quickly and succinctly describe what they had gotten out of their math courses. So I posed the question to a couple of other adults, one an educator and the other a graphic artist. The educator told me that she couldn't remember taking a math course as an undergraduate but she had taken a statistics course as part of her master's program. What she got out of it, she told me, was an A. Upon further probing, she said she didn't remember anything about the course except that she liked it and she thought she was required to take it because it would help her in reading research papers—something she never did.
The graphic artist told me that she had taken a math class and gotten a couple of running mates out of it. She liked to run and discovered that a couple of her classmates did also and after math class they went for a run. She, too, got a good grade out of the class and. when pressed, said she did get a couple of other things out of the class: She learned that she could succeed in a mathematics class and she also learned in mathematics you have postulates, theorems, and axioms and you proved things, but she couldn't remember any particular thing they proved.
That's not surprising. Not much of the mathematical content of a course—procedures, definitions, formulas, and the like—is going to be retained unless it's revisited every now and then, no matter how it's taught. Over time, human beings tend to forget seemingly inconsequential details—and it's a good thing we do, otherwise our minds would be cluttered with vast amounts of information no longer of significance or use to us. We do have long term memories. How material gets there isn't precisely known but according to one psychiatrist, what events and information are retained is dependent upon "love for the subject matter and it's dramatic, emotional, auditory, and visual impact"—not exactly associations many people make with the quadratic formula.
Rather than spending time “thinking about how to help students retain more of what they study for a longer period of time, “I suggest we think about something else: how to provide our students with a mathematical experience that, if not memorable, is at least engaging. An experience that recognizes their mathematical potential, evokes their mathematical curiosity, and allows them to exercise and strengthen their capacity to deal with mathematical matters. They may not remember the law of cosines or the derivative of logx but they will develop their mathematical proficiency and confidence, so if they ever need or want to review a topic or expand their knowledge, they will be able and willing to do so.