Summary of Demby (1997)
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
1
Summary of Demby (1997)
2
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.
Summary of Demby (1997)
3
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)
Summary of Demby (1997)
4
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 eighthgrade 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.
Summary of Demby (1997)
5
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
Summary of Demby (1997)
6
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.