Posting a reading reflection I just submitted for my C&I course.
Schoenfeld (1988) argued in When Good Teaching Leads to Bad Results: The Disasters of “Well-Taught” Mathematics Courses that [the then current] mathematics curricula, and pedagogy, did not enable students to experience true understanding, and hence, “fails to connect their formal symbolic manipulation procedures with the ‘real-world’ objects represented by the [mathematical] symbols” as well as “the underlying mathematical ideas.”* He also quotes Doyle (1988), coincidentally published in the same journal as Schoenfeld’s article, as “[suggesting] that the presentation of subject matter [mathematics] as familiar work – routinized exercises that can be worked out of context and without significant understanding of the subject matter – can trivialize that subject matter and deprive students of the opportunity to understand and use what they have studied.” He goes on to mention that the [then prevalent] predominant model of current instruction, is inadequate; it was called “the absorption theory of learning” from Romberg and Carpenter (1986) who stated: “The traditional classroom focuses on competition, management, and group aptitudes; the mathematics taught is assumed to be a fixed body of knowledge, and it is taught under the assumption that learners absorb what has been covered.” Sadly, not much has changed in the intervening two plus decades, even with the benefits of resurgent reforms by constructivists known to the educational profession.
In essence, teaching then, and today, emphasizes algorithmic procedures divorced from deep understanding, leaving students floundering when faced with any problem that deviates in construction from those found in traditional texts, homework, class assignments, course assessments, and standardized tests. Its even worse in a sense, since student performance on standardized tests has fallen considerably over the past three decades, inspite of their supposed lack of mathematically challenging problems.
Nonetheless, I could not help feeling that Schoenfeld held too narrow, and perhaps too romantically nostalgic, of a perspective of what it means to learn mathematics, such as when he emphasized the “exhilarating” nature of sense-making he experienced in a class when he was a student; a nature of sense-making that, as he states, “should be part of the pleasure of learning mathematics.” While I do not deny Schoenfeld, or others, who revel in real and imaginary numbers, and their interrelationships, I do not believe the plurality of students share this passion, and hence, it should not serve as a driving force in educational reform.
Furthermore, Schoenfeld goes on to recall from his school years, and to comment on his recent observations of a classroom, that “There was little sense of exploration, or of the possibility that the students could make sense of the mathematics themselves.” In some ways, he comes across as very judgmental, and a bit acerbic, when he describes the teachers and students he observed, as well as the broader educational field to include schools, state departments of education, and testing services; perhaps some of his points are on target, especially the further removed from the classroom, however, the other factors impacting student performance do not seem to be accounted for in his analysis.
Also, while I agree that the preponderance of mathematical instruction, then, and today, involves a teacher imparting knowledge to students, a sudden shift to the other end of the spectrum could yield disastrous results; perhaps a transition to a more measured mixture of the different pedagogical approaches is in order – one similar to which Lobato, Clarke, and Ellis (2005) advocate. Additionally, for a variety of reasons, not all students will make sense out of every single topic presented to them in school; while we desire that all students do make sense of everything, and we will strive to create the space for everyone to do so, we must recognize that not every student will be so capable, or feel so compelled. Having said that, if students try their hardest, reach out for help, and receive supportive instruction in a student-centered learning environment, they will have the best chance of reaching their greatest potential.
At the same time, I could not agree more with Schoenfeld’s six assumptions, below. The challenge, I believe, is restructuring school so adequate time is available for students to learn, and perhaps to re-learn if prior learning is forgotten, the foundational mathematics necessary to succeed in manipulating numbers while striving to make sense of context-sensitive, real-world mathematics problems that engage students and make sure they develop the competencies needed to continue in their academic pursuits, whether it be towards a four-year college, two-year college, or career-technical-education path.
(1) a major purpose of mathematics instruction is to help students learn to think mathematically. To elaborate, my assumptions about mathematics are as follows:
(2) even at the most elementary levels, mathematics is a complex and highly structured subject;
(3) thinking mathematically consists not only of mastering various facts and procedures, but also in understanding connections among them; and
(4) thinking mathematically also consists of being able to apply one’s formal mathematical knowledge flexibly and meaningfully in situations for which the mathematics is appropriate. Finally, I make some (basically constructivist) assumptions about humans as learners:
(5) students are active interpreters of the world around them, constantly building interpretive frameworks to make sense of their experiences; and
(6) those interpretive frameworks shape the ways that students see the world and act in it — in particular, how they see and use their mathematical knowledge.
* This is not a new argument, per se, since Dewey (1933) and Piaget (1970) made related claims. As such, I wonder whether the continued, systemic nature of the challenges facing students of mathematics transcends curricula and pedagogy and rests in the rigid structure of insufficiently allocated time slots for teaching students coupled with district-wide, time-sequenced instruction, along with high-stakes standardized testing, rather than individualized, learning-sequenced instruction; somewhat akin to date-certain versus task-complete contract language. But that is for a future Ph.D. dissertation.