Tougaloo College Institute for Mathematics (TCIM)
Empirical Research Article
Generalization Strategies of Beginning High School Algebra Students
Rivera, F.D. & Becker, J. (2003). The effects of figural and numerical cues on the induction processes of pre service
elementary mathematics teachers. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceeding of the 2003 Joint Meeting
of PME and PMENA (pp. 4-63-70). Honolulu, Hawaii: University of Hawaii.
Introductory/Purpose/Objective/Research Question/Focus of Study
The research questions that were addressed were as follows: What enables/hinders students’ abilities to generalize
a linear pattern? What strategies do successful students use to develop an explicit generalization? How do students make use
of visual and numerical cues in developing a generalization? Do students use different representations equally? Can students
connect different representations of a pattern with fluency? What can we glean from student work that will inform and improve
Twenty-three different strategies were identified falling into three types, numerical, figural, and pragmatic, based on
students’ exhibited strategies, understanding of variables, and representational fluency.
Twenty-two ninth grade students in a beginning algebra class in a public school in an urban setting participated in the
study. Eleven were males and eleven were females. Students were given several questions pertaining to patterns. They were
interviewed for about 20-20 minutes and asked to read the problems and to think aloud as they solved the problems. They were
audio taped. A graduate student analyzed the tapes and several follow up discussions and cross checking were conducted.
There were twenty-three strategies used. Students used several strategies as they solved different problems.
Visual Grouping Strategy (S1)
When given a pattern, Edward, a student, used a strategy of counting each “arm” of the pattern and
then multiplying to get the total.
Visual Growth of Each Arm Strategy (S4)
Similar to S1, students used the additive instead of the multiplicative approach to get the total number of tiles.
Counting ELLs and Adding 4 Center Squares (S14)
Another student, Alajandro, saw four groups of three white tiles forming an L shape around the center black cross with
an additional 4 center white squares on each side.
Numerical Use of Finite Differences in Table Strategy (S2)
Rosendo used finite differences in a table.
Trial and Error Strategy (S6/S6’)
Some students used random trial and error, while others used systematic trial and error.
The students’ strategies were graphed on an Excel program to examine trends as the task changed from specific
to general. One student, Katrina, began the problem with visual strategies then changed to numerical use of finite differences
to get a general formula. Then, she used the formula to find the values for the 10th pattern. Rani began with a visual strategy
then transitioned into using finite differences.
In the part of the task that caused students to transition from specific to general, twelve students used Strategy #17
(get a formula and substitute to get 10th term), in addition to other strategies.
Inability to Generalize (S3)
The results on generalization of the 22 students are as follows: two of the 13 classified as unable to generalize had
no success on any part of the problem, while the rest were able to do the first three parts of the task. All but one of the
11 who were unable to generalize started with a visual strategy but transitioned to a numerical one. The most common numerical
strategy was to extend the table. One student confused the roles of the independent and dependent variables, and another left
out the independent variable.
The study was consistent with findings taken from an earlier study conducted with pre-service elementary teachers. Overall,
students’ strategies appeared to be mainly numerical. Three types of generalizations based on similarity (numerical,
figural, and pragmatic) were identified in this study. Students who use numerical generalization use trial and error as a
similarity strategy with no sense of what the coefficients in the linear pattern represent. Students who use figural generalization
use perceptual similarity strategies in which the focus is on relationships among numbers in the linear sequence. Students
who use pragmatic generalization use both numerical and figural strategies and are representation-ally fluent. They see sequences
of numbers as consisting of both properties and relationships.
Figural generalizes eventually become pragmatic. Students who fail to generalize tend to start out with numerical strategies,
but they lack the flexibility to try other approaches and see possible connections between different forms of representation
and generalization strategies.
My Reaction to the Study
I thought it was very interesting that the 22 students used 23 strategies in the process of completing their math problems.
I believe that everyone is unique and has their own special way of doing things. This study proved just that. Each student
used several strategies when finding the solutions to their problems. They each processed the information differently and
took different steps in solving the problems. Some used trial and error, others extended the table to find the solution, and
still others got a formula and substituted to get the 10th term. They each had their own way at arriving at the answer.
Teachers should learn a lot from this study. The 22 students in this study all had different ways of computing the information
to arrive at the answer. This proved that they each had a particular way of learning and processing information. Teachers
should use differentiated instruction to foster learning for the many different intelligences and learning styles. Not only
will this kind of teaching enable all students to learn the task at hand, but it will also allow students to know more than
one way to find a solution. With this information, they could check their answers.
My argument is that there may not have been enough research subjects or participants in this study in order to make a
reliable conclusion. A larger population would make the results more valid and reliable.