Summary of Demby (1997) 1
Running head: SUMMARY OF DEMBY (1997)
Analytic Summary of Demby (1997):
Algebraic Procedures used by 13-to-15-year-olds
Samuel Otten
Michigan State University
Analytic Summary of Demby(1997):
Algebraic Procedures used by 13-to-15-year-olds
For most students, the first mathematical excursion carrying them away from arithmetic is the journey into the world of algebra. Though it is not mentioned explicitly, the primary problem addressed by Demby’s article is that students make many errors in this transitional stage as they begin to learn algebraic procedures and perform algebraic manipulations. There was work from various parts of Europe cited to add credence to this problem, some of which empirically documented students’ troubles and some of which called for improvements in instruction. The author did not feel the need to make a case for why this is an important problem, most likely because this was already clear to the targeted audience.
The purpose of the study was to categorize and characterize procedures used by middle-school aged students to simplify algebraic expressions, thus adding to the overall theory of algebra learning. The research sought to determine whether the procedures were teacher- or student-generated, which clearly has didactical implications. Furthermore, the stability of the procedures over time was investigated, as was the correlation between certain procedures and overall performance. As will be seen below, Demby did an effective job in the introduction of piquing the interest of the reader with regard to the main points that recurred in the concluding discussion. There was, however, at least one purpose that became apparent in the conclusion that was absent from the introduction, this being the purpose of supporting claims made in some prior work while challenging claims made in other prior work.
Theoretical Perspective
The primary theoretical framing found in this article springs from a longstanding debate concerning algebra learning, namely, whether it should be taught as generalized arithmetic or as a symbolic system equipped with formal rules. The former is associated with semantics where some form of meaning is attached to algebraic objects and expressions. The latter is associated with syntactics where algebraic objects and expressions are operated upon according to first principles and derived rules.
Some framing also took place with regard to error theory. In this domain the language of several other authors was incorporated, such as degenerate formalism which refers to egregiously erroneous student behavior stemming from a meaningless manipulation of symbols. Demby also drew upon previous work that pointed out students’ tendencies to formulate their own, often invalid, algebraic rules which are based on associations, not meaning or deduction.
Research Questions
The research questions were listed on the first page of the article, which I found very helpful because they were made explicit at the onset and I could connect to them as I read each of the remaining sections. They were as follows:
What procedures are used by the students while performing such tasks? Are they mostly procedures that were explained and used earlier by the teacher? Or perhaps most procedures are spontaneous, that is, invented by children themselves? What can be said about the development of students' procedures? Are they stable over a long period? Is there a correlation between the types of procedures and the achievements of the students using them? (p. 45)
Method
Demby administered a survey to 100 seventh-grade students in Poland and then selected 50 students for a subsequent interview. This process was repeated with the same students and a new survey a year later. It should be noted that one-fourth of the study population were students of Demby herself, and they were all from the same school so it may be assumed that they are reasonably homogenous in terms of mathematical background and perhaps even cultural background. The accuracy of these assumptions is not addressed in the article.
Findings
The author was able to distinguish between seven types of student procedures: automatization (A), formulas (F), guessing-substituting (GS), preparatory modification (PM), concretization (C), rules (R), and quasi-rules (QR). Student procedures were found to evolve from seventh-grade to eighth-grade with (R) being the most common procedure at both times but more prevalent in the second test. More types of procedures were used on the eighth-grade test. The (QR) procedure existed at both times. Failing students usually employed incorrect (R) or (QR). Partially successful students usually used (R). Successful students used the widest variety of procedures, frequently (R) and something else. Almost all users of (GS), (C), and (PM) in seventh-grade were at least partially successful, and almost all of these same students had become successful by the eighth-grade test. The (C) procedure decreased over time while (PM) increased.
These findings were presented in a statistically elementary way. Demby simply used percentages and total occurrences to discuss trends. There was no argument made that these trends or results were statistically significant, though they seemed to be.
Conclusions
Based on the empirical evidence, three groups of students could be distinguished quantitatively (by their percentage of success) and also qualitatively (by the procedures and number of procedures used). There were many students who used incorrect rules in grade seven but improved to correct rules in grade eight. This led Demby to posit that the emergence of incorrect rules may be a normal stage in the development of math students. If these erroneous rules are considered and discussed with peers and teachers in a rich learning environment, the student may be able to progress to strong understanding of correct rules.
This work suggested that there is a need to discriminate between (QR) and incorrect (R), the former being a procedure composed of inconsistent applications of invented rules and the latter being a procedure composed of consistent applications of invalid rules. Users of incorrect (R) were able to become successful over time, but users of (QR) were much more likely to remain at a low performance level. This distinction is largely missing from previous work. Also in regard to previous work, this research confirmed that students invent their own procedures, (C) is often used though rarely taught, (F) is rarely used though often taught, and successful students go through overgeneralization before reaching fluency. This research challenged the notion of some other researchers that students have to understand the formal basis of algebraic algorithms to achieve success. Demby notes that success is most strongly correlated with diverse procedures and so argues that both semantic and syntactic understanding is desired. Thus, didactically, it is wrong for a teacher to give general rules without previous experience and it is wrong to practice without recapitulation. “The most important thing is that children construct rules themselves” (p. 68).
Overall Remarks
This study, with its fairly straightforward design, was able to speak to a wide variety of issues in mathematics education. It was able to do so by placing itself within a rich web of previous and ongoing work, for example algebraic error theory, algebraic procedural categorization, semantic and syntactic understanding, developmental learning theory, and pedagogical philosophy.
I found it interesting that the use of a variety of procedures was a better predictor of success than any particular procedure alone. I also found it interesting from a curricular point of view that formulaic procedures were almost never used by students, though in my experience they are common in textbooks not only as propositions, which is fitting, but as justification for examples, which now does not seem as fitting. Finally, the Polish author struck a chord with myself here in the United States when she lamented the fact that students often just want the rules (see Skemp, 1977), making it challenging for a teacher to develop the wide variety of procedural and conceptual skills that are desirable.
References
Demby, A. (1997). Algebraic procedures used by 13-to-15-year-olds. Educational Studies in Mathematics, 33(1), 45-70.
Skemp, R. R. (1977). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26.